
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
Alice and Bob play a game on a strip of squares with \( n \geq 3 \) squares and two game pieces. At the beginning, Alice's piece is on the first square and Bob's piece is on the last square. They take turns moving their pieces. A move consists of moving one's own piece by one or two squares towards the opponent's piece to an empty square, without jumping over the opponent's piece. Alice moves first. The player who cannot make a move loses.

For which \( n \) can Bob win for sure, no matter how Alice plays? For which \( n \) can Alice win for sure, no matter how Bob plays?

(Karl Czakler)

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