
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  $P$  with real coefficients such that   $$ \frac{P(x)}{yz}+\frac{P(y)}{zx}+\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x) $$   holds for all nonzero real numbers  $x,y,z$  satisfying  $2xyz=x+y+z$ .

*Proposed by Titu Andreescu and Gabriel Dospinescu*

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