
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, with proof, all nonconstant polynomials \( P(x) \) with real coefficients such that, for all nonzero real numbers \( z \) with \( P(z) \neq 0 \) and \( P\left(\frac{1}{z}\right) \neq 0 \), we have
$$
\frac{1}{P(z)}+\frac{1}{P\left(\frac{1}{z}\right)}=z+\frac{1}{z}.
$$

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