(NIMO 2014) Sum of Powers of i

Algebra Level 5

Let \(S = \left\{ 1,2, \dots, 2014 \right\}\). Suppose that \[ \sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi \] where \(p\) and \(q\) are integers, \(i = \sqrt{-1}\), and the summation runs over all \(2^{2014}\) subsets of \(S\). Find the remainder when \(\left\lvert p\right\rvert + \left\lvert q \right\rvert\) is divided by \(1000\). (Here \(\left\lvert X \right\rvert\) denotes the number of elements in a set \(X\).)

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