
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
In a pile with \( n \) stones, two players play a game. On each turn, a player can take from the pile either a number of stones equal to a prime divisor of the current number of stones in the pile or 1. The player who takes the last stone wins. For which values of \( n \) can the starting player always play in such a way that guarantees a win, no matter how the opponent plays?

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