
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
Let  $ABC$  be an arbitrary triangle. A regular  $n$ -gon is constructed outward on the three sides of  $\triangle ABC$ . Find all  $n$  such that the triangle formed by the three centres of the  $n$ -gons is equilateral.

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