
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
There were 20 people gathered in a castle, 11 of whom are wise men who always tell the truth, and the remaining 9 are jesters who always lie. Around a table, $n \geq 3$ people sat, each of whom stated, "Exactly one of my two neighbors is a jester." For which values of $n$ is this possible? (It is necessary to find all possible $n$ and explain why other values cannot be.)

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