
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, a magpie perches on each vertex. After being startled, all magpies fly away. Later, they return to these vertices, with exactly one magpie per vertex, but not necessarily to their original vertices. Find all positive integers $n$ such that there must exist 3 magpies whose initial and final vertices form a triangle that is either all acute, all right, or all obtuse angles.

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