
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  $f$  with real coefficients such that for all reals  $a,b,c$  such that  $ab+bc+ca = 0$  we have the following relations

\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]

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