
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 $a_1, a_2, c_1, b_2, c_2$ be real numbers such that for every real number $x$, we have
\[ x^5 - x^4 + x^3 - x^2 + x - 1 = (x^3 + a_1x^2 + a_2x + c_1)(x^2 + b_2x + c_2). \]
Compute $a_1c_1 + b_2c_2$.

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