
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
There is a bird at each vertex of a regular n-gon. Suddenly the birds all fly away and then return a minute later. After their return there is again one bird at each vertex, but not necessarily the same bird as before. For which n can we always find three birds such that the triangle formed by their initial positions is of the same type (acute, obtuse or right-angled) as the triangle formed by their final positions?

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