
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
In a regular $n$-gon, each vertex initially has one magpie sitting on it. After being startled, all the magpies fly away. After some time, the magpies return to the vertices, each vertex again having exactly one magpie, though not necessarily the same magpie as before. Find all positive integers $n$ such that there always exist 3 magpies, forming a triangle with their initial and current positions, which will either all be acute triangles, all be right triangles, or all be obtuse triangles.

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