# 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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