
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 $2n$-sided polygon inscribed in a circle $(n \geqslant 2)$, there is a frog sitting at each vertex. At a certain moment, all frogs simultaneously jump to an adjacent vertex (a vertex is allowed to have multiple frogs). This movement is called a "jump". Determine all possible values of $n$ such that for the polygon, there exists a jump after which the line segment connecting any two different vertices with frogs does not pass through the center of the circle.

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