
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
An  $ (n, k) \minus{}$  tournament is a contest with  $ n$  players held in  $ k$  rounds such that: $ (i)$  Each player plays in each round, and every two players meet at most once. $ (ii)$  If player  $ A$  meets player  $ B$  in round  $ i$ , player  $ C$  meets player  $ D$  in round  $ i$ , and player  $ A$  meets player  $ C$  in round  $ j$ , then player  $ B$  meets player  $ D$  in round  $ j$ .

Determine all pairs  $ (n, k)$  for which there exists an  $ (n, k) \minus{}$  tournament.

*Proposed by Carlos di Fiore, Argentina*

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