
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
Thirty people are seated around a circular table. Each person is either an idiot or a wise person. Each person is asked: "Is the person to your right wise or an idiot?" Wise people always give the correct answer, while idiots may answer correctly or incorrectly. Given that the number of idiots does not exceed \( F \), determine the maximum value of \( F \) such that it is always possible to identify at least one wise person.

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