
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 the function \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for all real numbers \( x \) and \( y \),
\[ 
f(xy)(x + f(y)) = x^2 f(y) + y^2 f(x).
\]
(2015, Croatian Mathematical Olympiad)

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