
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 \( n \) be a natural number. A volleyball team consisting of \( n \) women and \( n \) men lines up for a game. Each team member occupies one of the positions \( 1, 2, \ldots, 2n \), whereby exactly positions \( 1 \) and \( n+1 \) are outside the court. During the game, all team members rotate, switching from position \( i \) to position \( i+1 \) (or from \( 2n \) to \( 1 \). How many possibilities are there for the starting line-up so that there are always at least \( n-1 \) women on the pitch, no matter how often the team rotates?

Note: Two starting line-ups are different if at least one team member occupies a different position.

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