
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 football league adopting a double round-robin format (i.e., each pair of teams plays two matches), there are \( m \) teams participating. After the tournament concluded, a total of \( 9n^2 + 6n + 32 \) matches were played, where \( n \) is an integer. Determine the value of \( m \).

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