If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a [[set]] of [[real numbers]], indexed so that $a_1 < a_2 < a_3 < \cdots < a_n,$ its ''complex power sum'' is defined to be $a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty [[subset]]s of $\{1,2,\ldots,n\}.$ Given that $S_8 = - 176 - 64i$ and $ S_9 = p + qi,$ where $p$ and $q$ are integers, find $\left\lvert p\right\rvert + \left\lvert q\right\rvert.$