
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $X$ .)

*Proposed by David Altizio*

Present the answer in LaTex format: \boxed{Your answer}
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