
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 \((n, k)\)-tournament is a contest with \(n\) players held in \(k\) rounds such that:
1. Each player plays in each round, and every two players meet at most once.
2. 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 \((n, k)\)-tournament.

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