
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 a positive integer \( m \geqslant 17 \), there are \( 2m \) players participating in a round-robin tournament. In each round, the \( 2m \) players are divided into \( m \) pairs, and each pair competes. In the next round, the pairings are rearranged. The tournament lasts for \( 2m-1 \) rounds, ensuring that each player competes against every other player exactly once. Determine the smallest possible positive integer \( n \) such that after some \( n \) rounds of a feasible tournament program, for any set of four players, either all of them have never competed against each other, or they have competed in at least two matches in total.

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