
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 \( f: \mathbf{Q}^{+} \rightarrow \mathbf{Q}^{+} \) that satisfy
$$
f(x) + f(y) + 2xy f(xy) = \frac{f(xy)}{f(x+y)},
$$
where \(\mathbf{Q}^{+}\) denotes the set of positive rational numbers.

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