
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 the ten-sided regular polygon \( P_1P_2...P_{10} \), how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set \( \{P_1, P_2, \ldots, P_{10}\} \)?

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