
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

Find the smallest positive integer \( n \) such that in any coloring of the \( n \)-gon's vertices with three colors (red, yellow, blue), there exist four vertices of the same color that form the vertices of an isosceles trapezoid (a convex quadrilateral with one pair of parallel sides and the other pair of equal non-parallel sides).

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