This one is for Mi!

Algebra Level 5

If \(\{a_1,a_2,a_3,\ldots,a_n\}\) is a set of real numbers, placed so that \(a_1 < a_2 < a_3 < \cdots < a_n,\) its complex power sum is defined as \(a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,\) where \(i^2 = - 1.\) Let \(S_n\) be the sum of the complex power sums of all nonempty subsets of \(\{1,2,\ldots,n\}.\) Given that \(S_8 = - 176 - 64i\) and \(S_9 = p + qi,\) where \(p\) ,\(q\) \(\in\) \(\mathbb{Z}\), find \(|p| + |q|\).

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I am posting this ques. because I have lost the bet I had with shubhendra.
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