
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
Consider the following two-player game. Player 1 starts with a number, \( N \). He then subtracts a proper divisor of \( N \) from \( N \) and gives the result to player 2 (a proper divisor of \( N \) is a positive divisor of \( N \) that is not equal to 1 or \( N \)). Player 2 does the same thing with the number she gets from player 1, and gives the result back to player 1. The two players continue until a player is given a prime number or 1, at which point that player loses. For which values of \( N \) does player 1 have a winning strategy?

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