
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
We selected the vertices, the centroid, and the trisection points of the sides of an equilateral triangle. How many points can we keep among them such that no three of them form an equilateral triangle?

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