
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
Given an integer  $n>1$ , let  $S_n$  be the group of permutations of the numbers  $1,\;2,\;3,\;\ldots,\;n$ . Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group  $S_n$ . It is forbidden to select an element that has already been selected. The game ends when the selected elements generate the whole group  $S_n$ . The player who made the last move loses the game. The first move is made by A. Which player has a winning strategy?

*Proposed by Fedor Petrov, St. Petersburg State University.*

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