
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 the sum of all  $x$  from  $2$  to  $1000$  inclusive such that   $$ \prod_{n=2}^x \log_{n^n}(n+1)^{n+2} $$   is an integer.

*Proposed by Deyuan Li and Andrew Milas*

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