
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 polynomials \( P \in \mathbb{R}[X] \) such that for all real numbers \( a, b, c \) satisfying \( ab + bc + ca = 0 \), we have:

\[
P(a-b) + P(b-c) + P(c-a) = P(a+b+c)
\]

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