
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  $P(x)$  be a real quadratic trinomial, so that for all  $x\in \mathbb{R}$  the inequality  $P(x^3+x)\geq P(x^2+1)$  holds. Find the sum of the roots of  $P(x)$ .

*Proposed by A. Golovanov, M. Ivanov, K. Kokhas*

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