
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 positive integers \( n \) and \( k \) (where \( k \leq 2n - 1 \)), there are \( 2n \) ping-pong players who played \( k \) rounds of matches. In each round, the players are divided into \( n \) pairs, and each pair competes in a match. Two players can compete more than once in different rounds. Determine the largest positive integer \( m = f(n, k) \) such that, regardless of how the matches are arranged, it is guaranteed that there will be \( m \) players who have never played a match against each other.

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