
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
$$
\frac{P(x)}{yz} + \frac{P(y)}{zx} + \frac{P(z)}{xy} = P(x - y) + P(y - z) + P(z - x)
$$
for all nonzero real numbers \( x, y, z \) obeying \( 2xyz = x + y + z \).

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