
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 \(P(x)\) with real coefficients such that for all nonzero real numbers \(x\), \[P(x)+P\left(\frac1x\right)     =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\]

*Proposed by Holden Mui*

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