
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
Given a positive integer \( n \geq 2 \), two players, A and B, play a coin game: At the beginning, A chooses a permutation of \( 1, 2, \ldots, n \) and writes it in a row on a piece of paper. Then, B selects one of the numbers and places a coin on this number. Subsequently, A moves the coin to an adjacent position, followed by B moving the coin to an adjacent position, and this process continues in each round. For each integer \( k \) in \( 1, 2, \ldots, n \), starting from B's first move, the coin can be placed on the number \( k \) at most \( k \) times. The player who cannot move the coin loses. For which values of \( n \) does player A have a winning strategy?

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