
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 \( m \geq 17 \), there are \( 2m \) players competing in a round-robin tournament. Each round, the \( 2m \) players are divided into \( m \) pairs, with each pair playing one match. In the next round, the pairs are rearranged and the competition continues for a total of \( 2m - 1 \) rounds, ensuring that each player competes against every other player exactly once.

Find the smallest possible positive integer \( n \) such that, after some rounds, there exists a feasible schedule where, for any 4 players, they either have not played against each other at all or have played at least 2 matches in total among themselves.

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