
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 $b_1$, $b_2$, $b_3$, $b_4$, $c_1$, $c_2$, $c_3$, and $c_4$ be real numbers such that for every real number $x$, we have
\[
  x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3)(x^2 + b_4 x + c_4). 
\]
Compute $b_1 c_1 + b_2 c_2 + b_3 c_3 + b_4 c_4$.

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