
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 the cubic polynomial \( P(x) = ax^3 + bx^2 + cx + d \) have real coefficients. It satisfies the inequality for all real numbers \( x \):
\[ P(x^3 - x) \geq P(x^2 - 1) \]
Determine the sum of the roots of \( P(x) \).

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