
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
Two players take turns painting the sides of an $n$-gon. The first player can paint a side that is adjacent to zero or two painted sides, while the second player can paint a side that is adjacent to one painted side. The player who cannot make a move loses. For which values of $n$ can the second player win regardless of how the first player plays?

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