
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 \( n \) vertices of a regular \( n \)-gon with three colors (red, yellow, and blue), there exist four vertices of the same color forming an isosceles trapezoid. (An isosceles trapezoid is a convex quadrilateral with one pair of parallel sides and the other pair of non-parallel sides being equal in length.)

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