
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
There are \(12k\) people attending a meeting, and each person shakes hands with exactly \(3k+6\) other people. Furthermore, for any two individuals, the number of people who have shaken hands with both of these individuals is the same. How many people are attending the meeting? Provide proof for your conclusion.

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