
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
On the board, a positive integer \( N \) is written. Two players take turns. On their turn, a player either replaces the number on the board with one of its divisors (excluding 1 and the number itself) or decreases the number by one (provided the resulting number remains positive). The player who cannot make a move loses. For which values of \( N \) can the first player guarantee a win, regardless of the opponent's strategy?

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