
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 \( m \geq 2 \). A meeting has \( 3m \) representatives in attendance. Each pair of representatives either shakes hands once or not at all. For a positive integer \( n \) (where \( n \leq 3m - 1 \)), if there exist \( n \) representatives such that they have shaken hands 1, 2, ..., \( n \) times respectively, then the meeting is called " \( n \)-interesting." If for every possible \( n \)-interesting meeting, there always 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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