
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 functions  $f: \mathbb{R} \rightarrow \mathbb{R}$  that satisfy   $$ (f(a)-f(b))(f(b)-f(c))(f(c)-f(a)) = f(ab^2+bc^2+ca^2) - f(a^2b+b^2c+c^2a) $$  for all real numbers  $a$ ,  $b$ ,  $c$ .

*Proposed by Ankan Bhattacharya, USA*

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