
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 quadratic polynomial $P(x) = ax^2 + bx + c$ with real coefficients, satisfy
\[P(2x^5 + 3x) \ge P(3x^4 + 2x^2 + 1)\]
for all real numbers $x$. Find the sum of the roots of $P(x)$.

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