
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
Denote by \(\mathbb{Q}^{+}\) the set of all positive rational numbers. Determine all functions \(f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}\) which satisfy the following equation for all \(x, y \in \mathbb{Q}^{+}\):
$$
f\left(f(x)^{2} y\right) = x^{3} f(x y) .
$$

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