
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
510 people went to a bookstore to buy books. It is known that (1) each person bought 3 different books, and (2) any two people have at least one book in common. What is the minimum number of people who bought the most popular book? Explain the reasoning.

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