
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
Suppose \(a, b_{1}, b_{2}, \cdots, b_{n}, c_{1}, c_{2}, \cdots, c_{n} \in \mathbf{R}\) such that
\[ x^{2n} + a \sum_{i=1}^{2n-1} x^{i} + 1 = \prod_{i=1}^{n} \left(x^{2} + b_{i} x + c_{i}\right) \]
holds for all real numbers \(x\). Determine the values of \(c_{1}, c_{2}, \cdots, c_{n}\).

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