
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
Let \( n \) be a positive integer. Two players, \( A \) and \( B \), play a game where they take turns choosing a positive integer \( k \) such that \( k \leq n \). The rules of the game are as follows:
1. No player can choose a number that has already been chosen by either player in any previous round.
2. No player can choose any number that is adjacent to any number they have chosen in any previous round.
3. If all numbers are chosen, the game is a draw; otherwise, the player who cannot choose a number loses the game.

If player \( A \) starts first and both players use optimal strategies, determine the outcome of the game.

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