
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
Compute the sum of all positive integers \( n < 1000 \) for which \( a_{n} \) is a rational number, given the sequence \( \{a_{n}\} \) defined by \( a_{1} = 1 \) and for all integers \( n \geq 1 \):
$$
a_{n+1} = \frac{a_{n} \sqrt{n^{2} + n}}{\sqrt{n^{2} + n + 2 a_{n}^{2}}}.
$$

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