
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  $ n$  points arbitrarily in the plane  $ P_{1},P_{2},\ldots,P_{n},$  among them no three points are collinear. Each of  $ P_{i}$  ( $1\le i\le n$ ) is colored red or blue arbitrarily. Let  $ S$  be the set of triangles having  $ \{P_{1},P_{2},\ldots,P_{n}\}$  as vertices, and having the following property: for any two segments  $ P_{i}P_{j}$  and  $ P_{u}P_{v},$  the number of triangles having  $ P_{i}P_{j}$  as side and the number of triangles having  $ P_{u}P_{v}$  as side are the same in  $ S.$  Find the least  $ n$  such that  in  $ S$  there exist two triangles, the vertices of each triangle having the same color.

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