
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,k$  be given positive integers satisfying  $ k\le 2n \minus{} 1$ . On a table tennis tournament  $ 2n$  players take part, they play a total of  $ k$  rounds match, each round is divided into  $ n$  groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer  $ m \equal{} f(n,k)$  such that no matter how the tournament  processes, we always find  $ m$  players each of pair of which didn't match each other.

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