
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
Given any $n$ points on a plane $p_{1}, p_{2}, \cdots, p_{n}$, where no three points are collinear, and each point $p_{i}(i=1,2,\cdots,n)$ is colored either red or blue. Let $S$ be a set of some triangles with vertex sets $\left\{p_{1}, p_{2}, \cdots, p_{n}\right\}$, having the property that for any two line segments $p_{i} p_{j}$ and $p_{h} p_{k}$, the number of triangles in $S$ with $p_{i} p_{j}$ as a side is equal to the number of triangles with $p_{h} p_{k}$ as a side. Find the smallest positive integer $n$ such that there are always two triangles in the set $S$ where each triangle's three vertices are the same color.

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