
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
18 soccer teams participate in a round-robin tournament, where in each round the 18 teams are divided into 9 groups, with each group consisting of two teams that play a match. Teams are reshuffled into new groups for each subsequent round, and the tournament includes a total of 17 rounds, ensuring that every team plays exactly one match against each of the other 17 teams. After playing $n$ rounds according to some arbitrary schedule, there always exist 4 teams that have played exactly 1 match among them. Find the maximum possible value of $n$.

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