
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 \( b_1 = b_2 = b_3 = 2 \). For \( n > 3 \), let \( b_n \) be the number of real numbers \( y \) such that 
\[
y^3 - 3b_{n-1}y + b_{n-2}b_{n-3} = 0.
\]
Compute the sum \( b_1 + b_2 + b_3 + \dots + b_{1000} \).

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