
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
Find all pairs  $(a, b)$  of coprime positive integers, such that  $a<b$  and   $$ b \mid (n+2)a^{n+1002}-(n+1)a^{n+1001}-na^{n+1000} $$   for all positive integers  $n$ .

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