
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 are playing a game on a strip of $n \geq 3$ squares with two game pieces. Initially, Alice's piece is on the first square, and Bob's piece is on the last square. The picture shows the starting position for a strip of $n=7$ squares. Moves are made alternately. A move consists of moving one's piece one or two squares towards the opponent's piece onto an empty square, without jumping over the opponent's piece. Alice moves first with her piece. A player loses if they can no longer make a move.

For which values of $n$ can Bob be guaranteed to win, regardless of how Alice plays? For which values of $n$ can Alice be guaranteed to win, regardless of how Bob plays?

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