
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 any real numbers \( x \) and \( y \), the following holds:
$$
f\left(x^{2}\right) f\left(y^{2}\right) + |x| f\left(-x y^{2}\right) = 3|y| f\left(x^{2} y\right).
$$

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