
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 \( m \geq 3 \). There are \( 2m \) players participating in a round-robin tournament. In each round, the \( 2m \) players are divided into \( m \) groups, with each group consisting of two players who compete against each other. The groups are reshuffled in each subsequent round, and a total of \( 2m-1 \) rounds are held, ensuring that each player competes against every other player exactly once. Find the smallest positive integer \( n \) such that there exists a feasible schedule of \( n \) rounds, where for any four players, either none of them have 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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