
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
Find all real-coefficient polynomials $P(x)$ such that for all real numbers $a, b, c$ satisfying $a b + b c + c a = 0$, the equality $P(a - b) + P(b - c) + P(c - a) = 2P(a + b + c)$ holds.

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