
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
The sequence $1001, 1004, 1009$ has a general term $a_{n} = n^{2} + 1000$, where $n \in \mathbf{N}_{+}$. For each $n$, let $d_{n}$ denote the greatest common divisor (GCD) of $a_{n}$ and $a_{n+1}$. Determine the maximum value of $d_{n}$ as $n$ ranges over all positive integers.

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