
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 an unequal-sided triangle on a plane, Peter and Basil take turns performing the following operation: each round, Peter selects a point on the plane, and Basil decides whether to color that point red or blue. Peter wins if there exists a triangle similar to the original triangle such that all its vertices are of the same color. What is the minimum number of rounds needed for Peter to ensure his victory regardless of Basil's strategy? (This value should be independent of the shape of the original triangle.)

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