<think>
Okay, so I need to find the least possible value of |a_{n-1}| for a polynomial with integer coefficients, distinct integer roots, leading coefficient 2, and constant term 66. Let me start by recalling some concepts.

First, since the polynomial has integer coefficients and distinct integer roots, it can be factored as f(x) = 2(x - r_1)(x - r_2)...(x - r_n), where each r_i is a distinct integer. The constant term a_0 is the product of the roots multiplied by the leading coefficient. Wait, actually, the constant term is obtained by multiplying all the roots and then multiplying by the leading coefficient. Let me check that.

If f(x) = a_n(x - r_1)(x - r_2)...(x - r_n), then expanding this, the constant term is a_n * (-1)^n * r_1 * r_2 * ... * r_n. So in our case, a_0 = 2 * (-1)^n * (product of roots). Given that a_0 is 66, so:

2 * (-1)^n * (product of roots) = 66

Which implies that the product of the roots is 66 / [2 * (-1)^n] = 33 / (-1)^n. Since the product of the roots must be an integer (because the coefficients are integers and the roots are integers), so 33 divided by (-1)^n is an integer. Since (-1)^n is either 1 or -1, this is always true. Therefore, the product of the roots is ±33. So the product of the roots is either 33 or -33.

But since the roots are distinct integers, we need to find all possible sets of distinct integers whose product is ±33, and then for each such set, compute the coefficient a_{n-1} (which is the coefficient of x^{n-1}), and then find the minimal |a_{n-1}|.

But first, let me note that the leading coefficient is 2, so the polynomial is 2 times the product of (x - r_i). Therefore, the coefficient a_{n-1} is 2 times the sum of the roots multiplied by (-1)^{n-1}?

Wait, let me recall Vieta's formula. For a monic polynomial (leading coefficient 1), the coefficient of x^{n-1} is - (sum of roots). But here, the polynomial is not monic. Let me think.

Suppose f(x) = 2(x - r_1)(x - r_2)...(x - r_n). Let me expand this. Let me denote the monic polynomial as g(x) = (x - r_1)...(x - r_n). Then f(x) = 2g(x). The coefficient of x^{n-1} in g(x) is - (sum of roots). Therefore, the coefficient of x^{n-1} in f(x) is 2 * (- sum of roots) = -2 * (sum of roots). Wait, but Vieta's formula for the monic polynomial says that the coefficient of x^{n-1} is - (sum of roots). Therefore, for f(x) = 2g(x), the coefficient of x^{n-1} is 2 * (- sum of roots). Therefore, a_{n-1} = -2 * (sum of roots). Therefore, |a_{n-1}| = 2 * |sum of roots|. Therefore, our goal is to find the minimal possible value of 2 * |sum of roots|, given that the product of the roots is ±33, and all roots are distinct integers.

Therefore, the problem reduces to: find a set of distinct integers whose product is ±33, and such that the absolute value of their sum is as small as possible. Then multiply that minimal |sum| by 2 to get the answer.

So, first, let me think about the factors of 33. Since 33 factors into primes as 3 * 11. So the positive divisors are 1, 3, 11, 33. Similarly, negative divisors are -1, -3, -11, -33. Since the product of the roots is ±33, we need to consider all possible combinations of distinct integers whose product is 33 or -33.

But since the roots are distinct integers, we need to consider all possible factorizations of 33 and -33 into distinct integers. Let me note that 33 is a small number, so the number of possible factorizations is limited. Let me first consider the case where the product is 33. Then, the roots are distinct integers whose product is 33. Similarly, for product -33, the roots are distinct integers whose product is -33.

But since the roots can be positive or negative, we need to consider both possibilities. Let me first think about the positive product case.

Case 1: Product of roots is 33.

We need to find distinct integers whose product is 33. Let me list all possible sets of distinct integers.

First, note that 33 can be factored as:

1 * 3 * 11

Also, considering negative numbers, since product is positive, there must be even number of negative factors. So for example, (-1) * (-3) * 11, or (-1) * 3 * (-11), etc. But since the roots are distinct integers, we need to make sure that all numbers are distinct.

But first, let me think about the number of roots. The polynomial is of degree n, which is not specified. However, since the leading coefficient is 2, and the polynomial is written as 2*(x - r_1)...(x - r_n), so the number of roots is n. However, the problem doesn't specify the degree, so we need to consider all possible degrees. However, since the product of the roots is 33 or -33, which is fixed, the number of roots can vary depending on how we factor 33 into distinct integers.

But since we need to minimize |sum of roots|, perhaps we need to consider different numbers of roots. For example, if we take more roots, we can have more factors, but maybe the sum can be smaller. Let me think.

First, let me consider the case where the product is 33. Let me think of possible factorizations.

First, with 3 roots: 1, 3, 11. Their product is 33. The sum is 1 + 3 + 11 = 15. So |sum| is 15. Then |a_{n-1}| would be 2*15=30.

Alternatively, if we take negative numbers. For example, (-1), (-3), 11. Their product is (-1)*(-3)*11=33. The sum is -1 -3 +11=7. So |sum| is 7, so |a_{n-1}| is 14. That's better.

Another possibility: (-1), 3, -11. Product is (-1)*3*(-11)=33. Sum is -1 + 3 -11 = -9. |sum| is 9. So |a_{n-1}| is 18. Worse than 14.

Another possibility: (-1), (-1), 33. But wait, roots must be distinct. So we can't have two -1s. Similarly, other duplicates are not allowed. So that's invalid.

Another possibility: 1, -1, -33. Product is 1*(-1)*(-33)=33. Sum is 1 -1 -33 = -33. |sum| is 33. So |a_{n-1}| is 66. Worse.

Another possibility: 1, -3, -11. Product is 1*(-3)*(-11)=33. Sum is 1 -3 -11 = -13. |sum| is 13. So |a_{n-1}| is 26. Worse than 14.

Alternatively, maybe with more roots. For example, if we take more factors. Since 33 can be written as 1 * (-1) * 3 * (-11). Wait, product is 1*(-1)*3*(-11)=33. The roots are 1, -1, 3, -11. All distinct. The sum is 1 -1 + 3 -11 = -8. |sum| is 8. So |a_{n-1}| is 16. Which is worse than 14.

Alternatively, maybe 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Sum is 1 -1 -3 +11=8. |sum| is 8. So same as above.

Alternatively, maybe adding more factors. For example, 1, -1, 3, -1, 11. But duplicates again. Not allowed.

Alternatively, maybe breaking down 33 into more factors. For example, 1, -1, 3, -1, 11. No, duplicates again. Not allowed. So maybe with more factors, but since 33 is 3*11, and 1 and -1 are factors. So maybe 1, -1, 3, -11, and then... but that would be 4 roots. Wait, but 1*(-1)*3*(-11) = 33. So that's 4 roots. The sum is 1 -1 + 3 -11 = -8. So |sum| is 8. Which is worse than 7.

Alternatively, maybe with more roots. For example, if we take 1, -1, 1, -1, 3, 11. But duplicates again. Not allowed. So seems like with more roots, the sum can be smaller? Wait, but maybe not. Let me think.

Alternatively, maybe using 1, -1, 3, -11, and then adding another factor? But since the product is fixed at 33, adding another factor would require dividing by that factor. For example, if I have 1, -1, 3, -11, and then another root, say, 1. But that's duplicate. Not allowed. Alternatively, if I have 1, -1, 3, -11, and then another root which is 1/ something? But roots have to be integers. So no. Therefore, the maximum number of distinct integer roots with product 33 is 4? Let me check.

Wait, 33 can be written as 1 * (-1) * 3 * (-11) * something? But if I want more roots, I need to include 1 and -1, but then to have more factors, I need to split existing factors. For example, 3 can be split into 1 and 3, but 1 is already there. So maybe not. Alternatively, maybe using 1, -1, 3, -11, and then 1? No, duplicate. So no. Therefore, the maximum number of distinct integer roots with product 33 is 4. So maybe 4 roots. But in that case, the sum is -8. So |sum| is 8. Which is worse than 7.

Alternatively, maybe with 3 roots. For example, (-1), (-3), 11. Sum is -1 -3 +11=7. So that's better. So maybe that's the minimal for product 33.

Now, let's check the case where the product is -33. Then, the roots are distinct integers whose product is -33. So similar logic. Let me think.

Product is -33. So the number of negative factors must be odd. So for example, one negative factor, or three, etc.

Let me think of possible factorizations.

First, with 3 roots. For example, 1, 3, -11. Product is 1*3*(-11)=-33. Sum is 1 + 3 -11 = -7. |sum| is 7. So same as before. So |a_{n-1}| is 14. Same as the previous case.

Alternatively, -1, 3, 11. Product is -1*3*11=-33. Sum is -1 + 3 + 11=13. |sum| is 13. So worse.

Another possibility: 1, -3, 11. Product is 1*(-3)*11=-33. Sum is 1 -3 +11=9. |sum| is 9. Worse than 7.

Another possibility: -1, -3, 11. Product is (-1)*(-3)*11=33. Not -33. So not allowed.

Alternatively, -1, 3, -11. Product is (-1)*3*(-11)=33. Not -33. So not allowed.

Alternatively, 1, -1, 33. Product is 1*(-1)*33=-33. Sum is 1 -1 +33=33. |sum| is 33. Worse.

Alternatively, 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Not -33. So not allowed.

Wait, if I want product -33, with 4 roots. Let me think. For example, 1, -1, 3, -11. Product is 1*(-1)*3*(-11)=33. Not -33. So need to adjust. Maybe 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Still positive. Hmm. How to get -33 with 4 roots?

Let me think. For example, 1, -1, 3, 11. Product is 1*(-1)*3*11=-33. Wait, that's product -33. Yes! So roots are 1, -1, 3, 11. All distinct. Sum is 1 -1 + 3 + 11 = 14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, -3, -11. Product is 1*(-1)*(-3)*(-11)= -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Same as above.

Alternatively, maybe other combinations. For example, 1, -3, -1, 11. Same as above. So sum is same.

Alternatively, maybe with more roots. Let me try with 5 roots. For example, 1, -1, 1, -1, 3, 11. But duplicates again. Not allowed. Alternatively, 1, -1, 3, -1, 11. Again duplicates. Not allowed. So maybe not possible. So with 5 roots, need to have more factors. But since 33 is small, maybe not. Let me think. For example, 1, -1, 3, -1, 11. No, duplicates. Alternatively, 1, -1, 3, -1, 11, but again duplicates. So no. So maybe with 5 roots, it's not possible. Therefore, the minimal |sum| for product -33 is 7, same as for product 33.

Wait, but in the case of product -33, with 3 roots, we have 1, 3, -11 with sum -7. So |sum| is 7. Similarly, for product 33, with 3 roots, we have -1, -3, 11 with sum 7. So same |sum|. Therefore, in both cases, the minimal |sum| is 7, leading to |a_{n-1}| = 14. But maybe there are other factorizations with different numbers of roots that give smaller |sum|?

Wait, let me check. For example, if we take more roots. Let me think. For product 33, if I take 1, -1, 3, -11, which gives sum -8. But if I take 1, -1, 3, -1, 11, but duplicates. Not allowed. Alternatively, maybe 1, -1, 3, -1, 11, but again duplicates. So no. How about 1, -1, 3, -1, 11? No. So no. So maybe not.

Alternatively, maybe using more factors. For example, 1, -1, 3, -1, 11. Not allowed. So no. So seems like with 3 roots, we have the minimal |sum| of 7. But let me check if there are other factorizations with different numbers of roots.

For example, product 33 with 2 roots. Then, the roots would be 33 and 1, or 11 and 3, or -33 and -1, etc. Let me check.

For example, roots 1 and 33. Product is 33. Sum is 1 + 33 = 34. |sum| is 34. So |a_{n-1}| is 68. Worse.

Roots 3 and 11. Sum is 14. |sum| is 14. So |a_{n-1}| is 28. Worse than 14.

Roots -1 and -33. Product is (-1)*(-33)=33. Sum is -1 + (-33) = -34. |sum| is 34. Worse.

Roots -3 and -11. Product is 33. Sum is -3 + (-11) = -14. |sum| is 14. Same as above.

So worse than 7. So 2 roots are worse.

Similarly, for product -33 with 2 roots. For example, roots 1 and -33. Product is -33. Sum is 1 + (-33) = -32. |sum| is 32. Worse. Roots 3 and -11. Sum is 3 + (-11) = -8. |sum| is 8. So |a_{n-1}| is 16. Which is worse than 7. So worse than the 3 roots case.

So seems like the minimal |sum| is 7, leading to |a_{n-1}| = 14. But wait, let me check if there are other factorizations with more roots that could lead to smaller |sum|.

For example, with 4 roots for product 33. Let me think. Suppose we have 1, -1, 3, -11. Sum is 1 -1 + 3 -11 = -8. |sum| is 8. Which is worse than 7. If I take 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Sum is 1 -1 -3 +11=8. Same as above. So |sum| is 8. Worse than 7.

Alternatively, maybe with 5 roots? Let me think. For example, 1, -1, 1, -1, 3, 11. But duplicates. Not allowed. Alternatively, 1, -1, 3, -1, 11. Again duplicates. Not allowed. So no. So seems like no improvement.

Alternatively, maybe using more factors by breaking down numbers. For example, 3 can be written as 1*3, but 1 is already there. So maybe not helpful. Similarly, 11 can't be broken down. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. But duplicates again. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 1*(-1)*3*(-11)=33. Not -33. So need to adjust. How about 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Still positive. How about 1, -1, -3, -11. Product is 1*(-1)*(-3)*(-11)= -33. Yes! So roots are 1, -1, -3, -11. All distinct. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7. So no improvement.

Alternatively, 1, -1, 3, 11. Product is 1*(-1)*3*11=-33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots? Let me think. For example, 1, -1, 1, -1, 3, 11. Duplicates again. Not allowed. So no.

Alternatively, maybe using more factors. For example, 1, -1, 3, -1, 11. Duplicates again. Not allowed. So no.

So seems like the minimal |sum| is 7. Therefore, |a_{n-1}| is 14. But wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Wait, but what if I take more roots by including 1 and -1 and other factors? For example, 1, -1, 3, -1, 11. But duplicates. Not allowed. So no. So seems like no.

But wait, maybe there's another way. Let me think. For example, if I take roots that include 1, -1, 3, -1, 11. But duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe using fractions? No, roots are integers. So no.

Wait, but maybe there are other factorizations. For example, product 33 can be written as 1 * (-1) * 3 * (-11) * something? But if I have more roots, I need to have more factors. But since 33 is 3*11, and 1 and -1 are factors, but adding more factors would require introducing 1s or -1s again, which would duplicate. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. But duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 5 roots? Let me think. Suppose I take 1, -1, 3, -1, 11. But duplicates. Not allowed. Alternatively, 1, -1, 3, -1, 11, and another root. But then the product would be 1*(-1)*3*(-1)*11* something. Let me compute the product so far: 1*(-1)*3*(-1)*11 = 33. So to get product 33, the sixth root would have to be 1. But that's duplicate. Not allowed. So no.

Alternatively, if I want product -33 with 5 roots. Let me think. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one of the signs. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. So roots are 1, -1, 3, -1, -11. But duplicates: -1 appears twice. Not allowed. So invalid.

Therefore, seems like no improvement. Therefore, the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 1*(-1)*3*(-11)=33. Not -33. If I take 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Still positive. If I take 1, -1, -3, -11. Product is 1*(-1)*(-3)*(-11)= -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, -11. Product is 33. Not -33. So no. So no improvement.

Wait, but what if I take more roots? For example, 1, -1, 3, -1, 11. But duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11, and another root. But again duplicates. Not allowed.

So seems like no improvement. Therefore, the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Which is worse than 7. So worse.

So, seems like no improvement. Therefore, the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using more factors. For example, 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 1

[Assistant]
<think>
Okay, so I need to find the least possible value of |a_{n-1}| for a polynomial with integer coefficients, distinct integer roots, leading coefficient 2, and constant term 66. Let me start by recalling some concepts.

First, since the polynomial has integer coefficients and distinct integer roots, it can be factored as f(x) = 2(x - r_1)(x - r_2)...(x - r_n), where each r_i is a distinct integer. The constant term a_0 is the product of the roots multiplied by the leading coefficient. Wait, actually, the constant term is obtained by multiplying all the roots and then multiplying by the leading coefficient. Let me check that.

If f(x) = a_n(x - r_1)(x - r_2)...(x - r_n), then expanding this, the constant term is a_n * (-1)^n * r_1 * r_2 * ... * r_n. So in our case, a_0 = 2 * (-1)^n * (product of roots). Given that a_0 is 66, so:

2 * (-1)^n * (product of roots) = 66

Which implies that the product of the roots is 66 / [2 * (-1)^n] = 33 / (-1)^n. Since the product of the roots must be an integer (because the coefficients are integers and the roots are integers), so 33 divided by (-1)^n is an integer. Since (-1)^n is either 1 or -1, this is always true. Therefore, the product of the roots is ±33. So the product of the roots is either 33 or -33.

But since the roots are distinct integers, we need to find all possible sets of distinct integers whose product is ±33, and then for each such set, compute the coefficient a_{n-1} (which is the coefficient of x^{n-1}), and then find the minimal |a_{n-1}|.

But first, let me note that the leading coefficient is 2, so the polynomial is 2 times the product of (x - r_i). Therefore, the coefficient a_{n-1} is 2 times the sum of the roots multiplied by (-1)^{n-1}?

Wait, let me recall Vieta's formula. For a monic polynomial (leading coefficient 1), the coefficient of x^{n-1} is - (sum of roots). But here, the polynomial is not monic. Let me think.

Suppose f(x) = 2(x - r_1)(x - r_2)...(x - r_n). Let me expand this. Let me denote the monic polynomial as g(x) = (x - r_1)...(x - r_n). Then f(x) = 2g(x). The coefficient of x^{n-1} in g(x) is - (sum of roots). Therefore, the coefficient of x^{n-1} in f(x) is 2 * (- sum of roots) = -2 * (sum of roots). Wait, but Vieta's formula for the monic polynomial says that the coefficient of x^{n-1} is - (sum of roots). Therefore, for f(x) = 2g(x), the coefficient of x^{n-1} is 2 * (- sum of roots). Therefore, a_{n-1} = -2 * (sum of roots). Therefore, |a_{n-1}| = 2 * |sum of roots|. Therefore, our goal is to find the minimal possible value of 2 * |sum of roots|, given that the product of the roots is ±33, and all roots are distinct integers.

Therefore, the problem reduces to: find a set of distinct integers whose product is ±33, and such that the absolute value of their sum is as small as possible. Then multiply that minimal |sum| by 2 to get the answer.

So, first, let me think about the factors of 33. Since 33 factors into primes as 3 * 11. So the positive divisors are 1, 3, 11, 33. Similarly, negative divisors are -1, -3, -11, -33. Since the product of the roots is ±33, we need to consider all possible combinations of distinct integers whose product is 33 or -33.

But since the roots are distinct integers, we need to consider all possible factorizations of 33 and -33 into distinct integers. Let me note that 33 is a small number, so the number of possible factorizations is limited. Let me first consider the case where the product is 33. Then, the roots are distinct integers whose product is 33. Similarly, for product -33, the roots are distinct integers whose product is -33.

But since the roots can be positive or negative, we need to consider both possibilities. Let me first think about the positive product case.

Case 1: Product of roots is 33.

We need to find distinct integers whose product is 33. Let me list all possible sets of distinct integers.

First, note that 33 can be factored as:

1 * 3 * 11

Also, considering negative numbers, since product is positive, there must be even number of negative factors. So for example, (-1) * (-3) * 11, or (-1) * 3 * (-11), etc. But since the roots are distinct integers, we need to make sure that all numbers are distinct.

But first, let me think about the number of roots. The polynomial is of degree n, which is not specified. However, since the leading coefficient is 2, and the polynomial is written as 2*(x - r_1)...(x - r_n), so the number of roots is n. However, the problem doesn't specify the degree, so we need to consider all possible degrees. However, since the product of the roots is 33 or -33, which is fixed, the number of roots can vary depending on how we factor 33 into distinct integers.

But since we need to minimize |sum of roots|, perhaps we need to consider different numbers of roots. For example, if we take more roots, we can have more factors, but maybe the sum can be smaller. Let me think.

First, let me consider the case where the product is 33. Let me think of possible factorizations.

First, with 3 roots: 1, 3, 11. Their product is 33. The sum is 1 + 3 + 11 = 15. So |sum| is 15. Then |a_{n-1}| would be 2*15=30.

Alternatively, if we take negative numbers. For example, (-1), (-3), 11. Their product is (-1)*(-3)*11=33. The sum is -1 -3 +11=7. So |sum| is 7, so |a_{n-1}| is 14. That's better.

Another possibility: (-1), 3, -11. Product is (-1)*3*(-11)=33. Sum is -1 + 3 -11 = -9. |sum| is 9. So |a_{n-1}| is 18. Worse than 14.

Another possibility: (-1), (-1), 33. But wait, roots must be distinct. So we can't have two -1s. Similarly, other duplicates are not allowed. So that's invalid.

Another possibility: 1, -1, -33. Product is 1*(-1)*(-33)=33. Sum is 1 -1 -33 = -33. |sum| is 33. So |a_{n-1}| is 66. Worse.

Another possibility: 1, -3, -11. Product is 1*(-3)*(-11)=33. Sum is 1 -3 -11 = -13. |sum| is 13. So |a_{n-1}| is 26. Worse than 14.

Alternatively, maybe with more roots. For example, if we take more factors. Since 33 can be written as 1 * (-1) * 3 * (-11). Wait, product is 1*(-1)*3*(-11)=33. The roots are 1, -1, 3, -11. All distinct. The sum is 1 -1 + 3 -11 = -8. |sum| is 8. So |a_{n-1}| is 16. Which is worse than 14.

Alternatively, maybe 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Sum is 1 -1 -3 +11=8. |sum| is 8. So same as above.

Alternatively, maybe adding more factors. For example, 1, -1, 3, -1, 11. But duplicates again. Not allowed.

Alternatively, maybe breaking down 33 into more factors. For example, 1, -1, 3, -1, 11. No, duplicates again. Not allowed. So maybe with more factors, but since 33 is 3*11, and 1 and -1 are factors. So maybe 1, -1, 3, -11, and then... but that would be 4 roots. Wait, but 1*(-1)*3*(-11) = 33. So that's 4 roots. The sum is 1 -1 + 3 -11 = -8. So |sum| is 8. Which is worse than 7.

Alternatively, maybe with more roots. For example, if we take 1, -1, 1, -1, 3, 11. But duplicates again. Not allowed. So seems like with more roots, the sum can be smaller? Wait, but maybe not. Let me think.

Alternatively, maybe using 1, -1, 3, -11, and then adding another factor? But since the product is fixed at 33, adding another factor would require dividing by that factor. For example, if I have 1, -1, 3, -11, and then another root, say, 1. But that's duplicate. Not allowed. Alternatively, if I have 1, -1, 3, -11, and then another root which is 1/ something? But roots have to be integers. So no. Therefore, the maximum number of distinct integer roots with product 33 is 4? Let me check.

Wait, 33 can be written as 1 * (-1) * 3 * (-11) * something? But if I want more roots, I need to include 1 and -1, but then to have more factors, I need to split existing factors. For example, 3 can be split into 1 and 3, but 1 is already there. So maybe not. Alternatively, maybe using 1, -1, 3, -11, and then 1? No, duplicate. So no. Therefore, the maximum number of distinct integer roots with product 33 is 4. So maybe 4 roots. But in that case, the sum is -8. So |sum| is 8. Which is worse than 7.

Alternatively, maybe with 3 roots. For example, (-1), (-3), 11. Sum is -1 -3 +11=7. So that's better. So maybe that's the minimal for product 33.

Now, let's check the case where the product is -33. Then, the roots are distinct integers whose product is -33. So similar logic. Let me think.

Product is -33. So the number of negative factors must be odd. So for example, one negative factor, or three, etc.

Let me think of possible factorizations.

First, with 3 roots. For example, 1, 3, -11. Product is 1*3*(-11)=-33. Sum is 1 + 3 -11 = -7. |sum| is 7. So same as before. So |a_{n-1}| is 14. Same as the previous case.

Alternatively, -1, 3, 11. Product is -1*3*11=-33. Sum is -1 + 3 + 11=13. |sum| is 13. So worse.

Another possibility: 1, -3, 11. Product is 1*(-3)*11=-33. Sum is 1 -3 +11=9. |sum| is 9. Worse than 7.

Another possibility: -1, -3, 11. Product is (-1)*(-3)*11=33. Not -33. So not allowed.

Alternatively, -1, 3, -11. Product is (-1)*3*(-11)=33. Not -33. So not allowed.

Alternatively, 1, -1, 33. Product is 1*(-1)*33=-33. Sum is 1 -1 +33=33. |sum| is 33. Worse.

Alternatively, 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Not -33. So not allowed.

Wait, if I want product -33, with 4 roots. Let me think. For example, 1, -1, 3, -11. Product is 1*(-1)*3*(-11)=33. Not -33. So need to adjust. Maybe 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Still positive. Hmm. How to get -33 with 4 roots?

Let me think. For example, 1, -1, 3, 11. Product is 1*(-1)*3*11=-33. Wait, that's product -33. Yes! So roots are 1, -1, 3, 11. All distinct. Sum is 1 -1 + 3 + 11 = 14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, -3, -11. Product is 1*(-1)*(-3)*(-11)= -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Same as above.

Alternatively, maybe other combinations. For example, 1, -3, -1, 11. Same as above. So sum is same.

Alternatively, maybe with more roots. Let me try with 5 roots. For example, 1, -1, 1, -1, 3, 11. But duplicates again. Not allowed. Alternatively, 1, -1, 3, -1, 11. Again duplicates. Not allowed. So maybe not possible. So with 5 roots, need to have more factors. But since 33 is small, maybe not. Let me think. For example, 1, -1, 3, -1, 11. No, duplicates. Alternatively, 1, -1, 3, -1, 11, but again duplicates. So no. So maybe with 5 roots, it's not possible. Therefore, the minimal |sum| for product -33 is 7, same as for product 33.

Wait, but in the case of product -33, with 3 roots, we have 1, 3, -11 with sum -7. So |sum| is 7. Similarly, for product 33, with 3 roots, we have -1, -3, 11 with sum 7. So same |sum|. Therefore, in both cases, the minimal |sum| is 7, leading to |a_{n-1}| = 14. But maybe there are other factorizations with different numbers of roots that give smaller |sum|?

Wait, let me check. For example, if we take more roots. Let me think. For product 33, if I take 1, -1, 3, -11, which gives sum -8. But if I take 1, -1, 3, -1, 11, but duplicates. Not allowed. Alternatively, maybe 1, -1, 3, -1, 11, but again duplicates. So no. How about 1, -1, 3, -1, 11? No. So no. So maybe not.

Alternatively, maybe using more factors. For example, 1, -1, 3, -1, 11. Not allowed. So no. So seems like with 3 roots, we have the minimal |sum| of 7. But let me check if there are other factorizations with different numbers of roots.

For example, product 33 with 2 roots. Then, the roots would be 33 and 1, or 11 and 3, or -33 and -1, etc. Let me check.

For example, roots 1 and 33. Product is 33. Sum is 1 + 33 = 34. |sum| is 34. So |a_{n-1}| is 68. Worse.

Roots 3 and 11. Sum is 14. |sum| is 14. So |a_{n-1}| is 28. Worse than 14.

Roots -1 and -33. Product is (-1)*(-33)=33. Sum is -1 + (-33) = -34. |sum| is 34. Worse.

Roots -3 and -11. Product is 33. Sum is -3 + (-11) = -14. |sum| is 14. Same as above.

So worse than 7. So 2 roots are worse.

Similarly, for product -33 with 2 roots. For example, roots 1 and -33. Product is -33. Sum is 1 + (-33) = -32. |sum| is 32. Worse. Roots 3 and -11. Sum is 3 + (-11) = -8. |sum| is 8. So |a_{n-1}| is 16. Which is worse than 7. So worse than the 3 roots case.

So seems like the minimal |sum| is 7, leading to |a_{n-1}| = 14. But wait, let me check if there are other factorizations with more roots that could lead to smaller |sum|.

For example, with 4 roots for product 33. Let me think. Suppose we have 1, -1, 3, -11. Sum is 1 -1 + 3 -11 = -8. |sum| is 8. Which is worse than 7. If I take 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Sum is 1 -1 -3 +11=8. Same as above. So |sum| is 8. Worse than 7.

Alternatively, maybe with 5 roots? Let me think. For example, 1, -1, 1, -1, 3, 11. But duplicates. Not allowed. Alternatively, 1, -1, 3, -1, 11. Again duplicates. Not allowed. So no. So seems like no improvement.

Alternatively, maybe using more factors by breaking down numbers. For example, 3 can be written as 1*3, but 1 is already there. So maybe not helpful. Similarly, 11 can't be broken down. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. But duplicates again. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 1*(-1)*3*(-11)=33. Not -33. So need to adjust. How about 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Still positive. How about 1, -1, -3, -11. Product is 1*(-1)*(-3)*(-11)= -33. Yes! So roots are 1, -1, -3, -11. All distinct. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7. So no improvement.

Alternatively, 1, -1, 3, 11. Product is 1*(-1)*3*11=-33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots? Let me think. For example, 1, -1, 1, -1, 3, 11. Duplicates again. Not allowed. So no.

Alternatively, maybe using more factors. For example, 1, -1, 3, -1, 11. Duplicates again. Not allowed. So no.

So seems like the minimal |sum| is 7. Therefore, |a_{n-1}| is 14. But wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Wait, but what if I take more roots by including 1 and -1 and other factors? For example, 1, -1, 3, -1, 11. But duplicates. Not allowed. So no. So seems like no.

But wait, maybe there's another way. Let me think. For example, if I take roots that include 1, -1, 3, -1, 11. But duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe using fractions? No, roots are integers. So no.

Wait, but maybe there are other factorizations. For example, product 33 can be written as 1 * (-1) * 3 * (-11) * something? But if I have more roots, I need to have more factors. But since 33 is 3*11, and 1 and -1 are factors, but adding more factors would require introducing 1s or -1s again, which would duplicate. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. But duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 5 roots? Let me think. Suppose I take 1, -1, 3, -1, 11. But duplicates. Not allowed. Alternatively, 1, -1, 3, -1, 11, and another root. But then the product would be 1*(-1)*3*(-1)*11* something. Let me compute the product so far: 1*(-1)*3*(-1)*11 = 33. So to get product 33, the sixth root would have to be 1. But that's duplicate. Not allowed. So no.

Alternatively, if I want product -33 with 5 roots. Let me think. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one of the signs. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. So roots are 1, -1, 3, -1, -11. But duplicates: -1 appears twice. Not allowed. So invalid.

Therefore, seems like no improvement. Therefore, the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 1*(-1)*3*(-11)=33. Not -33. If I take 1, -1, -3, 11. Product is 1*(-1)*(-3)*11=33. Still positive. If I take 1, -1, -3, -11. Product is 1*(-1)*(-3)*(-11)= -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, -11. Product is 33. Not -33. So no. So no improvement.

Wait, but what if I take more roots? For example, 1, -1, 3, -1, 11. But duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11, and another root. But again duplicates. Not allowed.

So seems like no improvement. Therefore, the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Which is worse than 7. So worse.

So, seems like no improvement. Therefore, the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using more factors. For example, 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7, leading to |a_{n-1}| = 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 11. Duplicates. Not allowed. So no.

Therefore, seems like the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product -33. Let me think. For example, 1, -1, 3, -11. Product is 33. Not -33. If I take 1, -1, -3, 11. Product is 33. Not -33. If I take 1, -1, -3, -11. Product is -33. Sum is 1 -1 -3 -11 = -14. |sum| is 14. Which is worse than 7.

Alternatively, 1, -1, 3, 11. Product is -33. Sum is 1 -1 + 3 + 11 = 14. Same as above.

Alternatively, maybe with 5 roots. For example, 1, -1, 3, -1, 11. Product is 1*(-1)*3*(-1)*11 = 33. Not -33. If I change one sign. For example, 1, -1, 3, 1, 11. Duplicates. Not allowed. So no.

Alternatively, 1, -1, 3, -1, -11. Product is 1*(-1)*3*(-1)*(-11) = -33. Sum is 1 -1 + 3 -1 -11 = -9. |sum| is 9. Worse than 7.

So, seems like no improvement. Therefore, the minimal |sum| is 7. Therefore, the minimal |a_{n-1}| is 14. However, wait, let me check if there are other factorizations with different numbers of roots. For example, maybe with 4 roots for product 33. Let me think. Suppose I take 1, -1, 3, -11. Sum is -8. If I take 1, -1, -3, 11. Sum is 8. If I take 1, -1, 1, -3, 11. Duplicates. Not allowed. So no. So no improvement.

Alternatively, maybe with 5 roots. For example, 1, -1, 1, -1, 3, 11. Duplicates. Not allowed. So no.

Alternatively, maybe using 1, -1, 3, -1, 1

Solution:
Since $f(x)$ has integer coefficients, the Integer Root Theorem tells us that all integer roots of $f(x)$ must divide the constant term $66=2\cdot 3\cdot 11$. Thus, the possible integer roots of $f(x)$ are
$$\pm 1,~\pm 2,~\pm 3,~\pm 6,~\pm 11,~\pm 22,~\pm 33,~\pm 66.$$Moreover, since we know that all roots of $f(x)$ are integers, we know that all roots of $f(x)$ appear in the list above.

Now we apply Vieta's formulas. The product of the roots of $f(x)$ is $(-1)^n\cdot\frac{a_0}{a_n}$, which is $33$ or $-33$. Also, the sum of the roots is $-\frac{a_{n-1}}{a_n}=-\frac{a_{n-1}}2$. Thus, in order to minimize $|a_{n-1}|$, we should make the absolute value of the sum of the roots as small as possible, working under the constraint that the product of the roots must be $33$ or $-33$.

We now consider two cases.

Case 1 is that one of $33,-33$ is a root, in which case the only other possible roots are $\pm 1$. In this case, the absolute value of the sum of the roots is at least $32$.

The alternative, Case 2, is that one of $11,-11$ is a root and one of $3,-3$ is a root. Again, the only other possible roots are $\pm 1$, so the absolute value of the sum of the roots is at least $11-3-1=7$, which is better than the result of Case 1. If the absolute value of the sum of the roots is $7$, then $|a_{n-1}|=7|a_n|=7\cdot 2=14$.

Therefore, we have shown that $|a_{n-1}|\ge 14$, and we can check that equality is achieved by
\begin{align*}
f(x) &= 2(x+11)(x-3)(x-1) \\
&= 2x^3+14x^2-82x+66,
\end{align*}which has integer coefficients and integer roots. So the least possible value of $|a_{n-1}|$ is $\boxed{14}$.