
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 \( p(x) \in \mathbf{R}[x] \) such that for all \( x \neq 0 \), the following equation holds:
\[ p(x)^{2} + p\left(\frac{1}{x}\right)^{2} = p\left(x^{2}\right) p\left(\frac{1}{x^{2}}\right). \]

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