
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
In a competition there are  $18$  teams and in each round  $18$  teams are divided into  $9$  pairs where the  $9$  matches are played coincidentally. There are  $17$  rounds, so that each pair of teams play each other exactly once. After  $n$  rounds, there always exists  $4$  teams such that there was exactly one match played between these teams in those  $n$  rounds. Find the maximum value of  $n$ .

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