
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
A traveler arrived on an island inhabited by liars (L) and truth-tellers (P). Each L, when asked a question like "How many...?", gives a number that is either 2 more or 2 less than the correct answer, while each P answers correctly. The traveler met two inhabitants and asked each how many L and P live on the island. The first one answered: "If you do not count me, there are 1001 L and 1002 P," and the second one answered: "If you do not count me, there are 1000 L and 999 P." How many L and P are actually on the island? Who are the first and the second inhabitants of the island?

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