
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 a positive integer \( n \geq 3 \) and a set \( M \) with \( n \) elements, find the largest positive integer \( k \) such that there exists a collection \( \psi \) of \( k \) three-element subsets of \( M \), where the intersection of any two elements in \( \psi \) is non-empty.

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