
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 \(a, b, c, d\) be real numbers such that \(b-d \geq 5\) and all zeros \(x_{1}, x_{2}, x_{3},\) and \(x_{4}\) of the polynomial \(P(x) = x^{4} + a x^{3} + b x^{2} + c x + d\) are real. Find the smallest value the product \(\left(x_{1}^{2} + 1\right)\left(x_{2}^{2} + 1\right)\left(x_{3}^{2} + 1\right)\left(x_{4}^{2} + 1\right)\) can take.

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