
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
On the island of Misfortune, there live knights who always tell the truth and liars who always lie. One day, 2023 natives, among whom $N$ are liars, stood in a circle and each said, "Both of my neighbors are liars". How many different values can $N$ take?


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