
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
One evening, 21 people communicate by phone $n$ times. It is known that among them, there are $m$ ($m$ is an odd number) people $a_{1}, a_{2}, \cdots, a_{m}$ such that $a_{i}$ communicates with $a_{i+1}$ $\left(i=1,2, \cdots, m; a_{m+1}=a_{1}\right)$. If none of these 21 people had a three-way conversation, find the maximum value of $n$.

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