
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
A regular  $2015$ -simplex  $\mathcal P$  has  $2016$  vertices in  $2015$ -dimensional space such that the distances between every pair of vertices are equal. Let  $S$  be the set of points contained inside  $\mathcal P$  that are closer to its center than any of its vertices. The ratio of the volume of  $S$  to the volume of   $\mathcal P$  is  $\frac mn$ , where  $m$  and  $n$  are relatively prime positive integers. Find the remainder when  $m+n$  is divided by  $1000$ .

*Proposed by James Lin*

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