
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
Varya and Miron are playing the following game. There is a pile of $n$ stones on the table. Players take turns making moves, with Varya starting first. During a move, the player splits any pile that has more than one stone into several equal smaller piles. The player who cannot make a move (when, before their turn, each pile has exactly one stone) loses. For which values of $n$ will Varya win, and for which values will Miron win, assuming both players play optimally?

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