
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
At each vertex of a regular $n$-gon, there is a magpie. When startled, the magpies all fly away. After some time, they all return to these points, still with one magpie at each vertex, but not necessarily at their original vertices. Determine all positive integers $n$ such that there must exist three magpies, such that the triangles formed by their initial and final vertices are all either acute, right, or obtuse triangles.

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