
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
On a circle, \( n > 1 \) points are marked, called positions, dividing it into equal arcs. The positions are numbered clockwise from 0 to \( n-1 \). Vasya places a token on one of these positions. The following actions, called moves, are repeated an unlimited number of times: Petya names a natural number, and Vasya moves the token clockwise or counterclockwise by the number of positions specified by Petya. If at any point after a move the token lands on position number 0, Vasya loses, and Petya wins. For which values of \( n \) can Petya always win, regardless of Vasya's moves?

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