
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
Let \( n \geq 2 \) be an integer. Ariane and Bérénice play a game on the set of congruence classes modulo \( n \). At the beginning, the residue class 1 is written on a piece of paper. In each turn, the player whose turn it is replaces the current residue class \( x \) either with \( x+1 \) or \( 2x \). The players take turns with Ariane starting.

Ariane wins if the residue class 0 is reached during the game. Bérénice wins if she can prevent this from happening indefinitely.

Determine, depending on \( n \), which of the two players has a winning strategy.

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