Intro to Imaginary Numbers

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 $X$.)

(Source: NIMO 15)