
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  $n$  be a positive integer.  $S$  is a set of points such that the points in  $S$  are arranged in a regular  $2016$ -simplex grid, with an edge of the simplex having  $n$  points in  $S$ . (For example, the  $2$ -dimensional analog would have  $\dfrac{n(n+1)}{2}$  points arranged in an equilateral triangle grid). Each point in  $S$  is labeled with a real number such that the following conditions hold:

(a) Not all the points in  $S$  are labeled with  $0$ .

(b) If  $\ell$  is a line that is parallel to an edge of the simplex and that passes through at least one point in  $S$ , then the labels of all the points in  $S$  that are on  $\ell$  add to  $0$ .

(c) The labels of the points in  $S$  are symmetric along any such line  $\ell$ .

Find the smallest positive integer  $n$  such that this is possible.

Note: A regular  $2016$ -simplex has  $2017$  vertices in  $2016$ -dimensional space such that the distances between every pair of vertices are equal.

*Proposed by James Lin*

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