
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 \(p(x)\) with real coefficients such that
\[ p(a+b-2c) + p(b+c-2a) + p(c+a-2b) = 3p(a-b) + 3p(b-c) + 3p(c-a) \]
for all \(a, b, c \in \mathbb{R}\).

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