
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 are knights who always tell the truth and liars who always lie living on an island. The island's population is 1000 people, spread across 10 villages (each village has at least two people). One day, each islander claimed that all other villagers in their village are liars. How many liars live on the island? (Two villagers are considered to be from the same village if they live in the same village.)

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