
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 $b_1 = b_2 = b_3 = 1$. For $n > 3$, let $b_n$ be the number of positive integers $x$ such that
\[
x^4 - 3b_{n - 1}x^2 + 2b_{n - 2}b_{n - 3} = 0.
\]
Compute the sum $b_1 + b_2 + b_3 + \dots + b_{15}$.

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