
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 math competition: (1) there are \( n \) problems where \( n \geq 4 \); (2) each problem is solved by exactly 4 people; (3) for any two problems, exactly 1 person solved both. If the number of participants is greater than or equal to \( 4n \), find the minimum value of \( n \) such that there exists at least 1 person who solved all the problems.

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