
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
In a school, \(n\) clubs are to be formed, with \(n \geq 3\) and each with 3 members, such that for each pair of clubs there is exactly one student who is a member of both. 

a) Give an example of a distribution of 7 clubs that satisfies the mentioned conditions.

b) Verify that if a student belongs to 4 clubs, then they must belong to all the others.

c) Determine the minimum value of \(n\) such that for any set of clubs that satisfy these conditions, the presence of the same student in all of them is mandatory.

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