
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
Assume integer  $m \geq 2.$  There are  $3m$  people in a meeting, any two of them either shake hands with each other once or not.We call the meeting " $n$ -interesting", only if there exists  $n(n\leq 3m-1)$  people of them, the time everyone of whom shakes hands with other  $3m-1$  people is exactly  $1,2,\cdots,n,$  respectively. If in any " $n$ -interesting" meeting, there exists  $3$  people of them who shake hands with each other, find the minimum value of  $n.$ 

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