
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
We say that a pile is a set of four or more nuts. Two persons play the following game. They start with one pile of \( n \geq 4 \) nuts. During a move, a player takes one of the piles that they have and splits it into two non-empty subsets (these sets are not necessarily piles; they can contain an arbitrary number of nuts). If the player cannot move, he loses. For which values of \( n \) does the first player have a winning strategy?

Answer: The first player has a winning strategy when \( n \equiv 0,1,2 (\bmod 4) \); otherwise the second player has a winning strategy.

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