
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 \( p_{1}, p_{2}, \cdots, p_{n} \) in a plane, where no three points are collinear, each point \( p_{i} \) (for \( 1 \leqslant i \leqslant n \)) is colored either red or blue. Let \( S \) be the set of triangles with vertices \(\{ p_{1}, p_{2}, \cdots, p_{n} \} \) such that for any two line segments \( p_{i}p_{j} \) and \( p_{u}p_{v} \), the number of triangles in \( S \) having \( p_{i}p_{j} \) as a side is the same as the number of triangles having \( p_{u}p_{v} \) as a side. Find the minimum value of \( n \) such that \( S \) always contains at least two triangles, each with vertices of the same color.

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