
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
Given an integer $m \geqslant 2$. A meeting is attended by $3m$ representatives, where each pair of representatives either shakes hands once or does not shake hands at all. For a positive integer $n$ $(n \leqslant 3m - 1)$, if there exist $n$ representatives such that the number of times they have shaken hands are $1, 2, \cdots, n$, then the meeting is called "$n$-interesting". For all possible $n$-interesting meetings, it is guaranteed that there exist three representatives who have all shaken hands with each other. Find the smallest value of $n$.

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