
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
Every point on the circumference of a circle is colored either red, yellow, or blue, and all three colors are present on the circle. If we randomly select \( n \) points from the circle and there always exist three points that form an obtuse triangle with vertices of the same color, what is the minimum possible value of \( n \)?

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