
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 \( P \in \mathbb{R}[X] \) be a monic polynomial of degree 2. Suppose that \( P(1) \geq P(0) + 3 \), and that \( P \) has two real roots (not necessarily distinct) \( x_1 \) and \( x_2 \). Find the smallest possible value of \( \left(x_1^2 + 1\right)\left( x_2^2 + 1\right) \).

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