
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  $N\ge2$  be a fixed positive integer. There are  $2N$  people, numbered  $1,2,...,2N$ , participating in a tennis tournament.  For any two positive integers  $i,j$  with  $1\le i<j\le 2N$ , player  $i$  has a higher skill level than player  $j$ . Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among  $N$  courts, numbered  $1,2,...,N$ .  

During a round, each player plays against the other person assigned to his court (so that exactly one match takes place per court), and the player with higher skill wins the match (in other words, there are no upsets).  Afterwards, for  $i=2,3,...,N$ , the winner of court  $i$  moves to court  $i-1$  and the loser of court  $i$  stays on court  $i$ ; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court  $N$ .

Find all positive integers  $M$  such that, regardless of the initial pairing, the players  $2, 3, \ldots, N+1$  all change courts immediately after the  $M$ th round.

*Proposed by Ray Li*

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