
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
Let \( n (\geqslant 3) \) be a positive integer and \( M \) be an \( n \)-element set. Determine the largest positive integer \( k \) such that there exists a family of \( k \) distinct 3-element subsets of \( M \) in which the intersection of any two 3-element subsets is non-empty.

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