
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 functions \( f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+} \) such that for all positive real numbers \( x \) and \( y \), the following equation holds:
\[ x f\left(x^{2}\right) f(f(y)) + f(y f(x)) = f(x y)\left(f\left(f\left(x^{2}\right)\right) + f\left(f\left(y^{2}\right)\right)\right). \]

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