
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
Let \(\mathbb{Q}^{+}\) denote the set of positive rational numbers. Find, with proof, all functions \(f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}\) such that, for all positive rational numbers \(x\) and \(y\), we have
\[
f(x) = f(x + y) + f\left(x + x^2 f(y)\right).
\]

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