
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
On a plane, there are $\mathrm{n}$ points $\mathrm{P}_{1} , \mathrm{P}_{2} , \ldots , \mathrm{P}_{\mathrm{n}}$, where no three points are collinear. Each point $\mathrm{P}_{\mathrm{i}}$ ($1 \leq i \leq n$) is painted in one of two colors, red or blue. Let $S$ be the set of all triangles formed by the vertices $\left\{P_{1}, P_{2}, \ldots, P_{n}\right\}$, with the property that for any two line segments $\mathrm{P}_{\mathrm{i}} \mathrm{P}_{\mathrm{j}}$ and $\mathrm{P}_{\mathrm{u}} \mathrm{P}_{\mathrm{v}}$, the number of triangles in $S$ that have $\mathrm{P}_{\mathrm{i}} \mathrm{P}_{\mathrm{j}}$ as an edge is the same as the number of triangles that have $\mathrm{P}_{\mathrm{u}} \mathrm{P}_{\mathrm{v}}$ as an edge. Determine the smallest value of $\mathrm{n}$ such that there are always at least two triangles in $S$ with vertices all of the same color.

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