
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 $2n$-gon ($n \geqslant 2$) inscribed in a circle, there is a frog on each vertex. At a certain moment, all the frogs simultaneously jump to an adjacent vertex (it is allowed for multiple frogs to be on the same vertex). This is called a "jump." Find all possible values of $n$ such that for this $2n$-gon, there exists a jumping method such that after the jump, 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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