
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 taking turns to choose positive integers \( k \leq n \). The rules are:
1. A player cannot choose a number that has already been chosen by either player on any previous turn.
2. A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
3. The game is a draw if all numbers have been chosen; otherwise, the player who cannot choose a number anymore loses the game.

Player \( A \) takes the first turn. Determine the outcome of the game if both players use optimal strategies.

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