
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
Determine all real polynomials \(P\) that satisfy, for all real numbers \(a, b, c\),
\[ P(a+b-2c) + P(b+c-2a) + P(a+c-2b) = 3P(a-b) + 3P(b-c) + 3P(c-a). \]

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