
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 drawing diagonals in a regular $(2n+1)$-gon $(n>1)$. A diagonal is allowed to be drawn if it intersects (at internal points) with an even number of previously drawn diagonals (and has not been drawn before). The player who cannot make a move loses. Who will win with optimal play?

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