
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
A competition has \( n \geq 2 \) participants, where each participant's score each day forms the set \( \{1, 2, \cdots, n\} \). At the end of day \( k \), if the sum of the scores of any two participants is exactly 52, find all pairs \( (n, k) \) that make this possible.

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