
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
Let \(a_1 = a_2 = a_3 = 1.\) For \(n > 3\), let \(a_n\) be the number of real roots \(x\) such that
\[
x^3 - 2a_{n-1}x + a_{n-2}a_{n-3} = 0.
\]
Compute the sum \(a_1 + a_2 + a_3 + \dots + a_{1000}.\)

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