
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 \(\mathbb{R}^{+}\) be the set of positive real numbers. Find all functions \(f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) such that \(f(x+f(y)) = y f(xy+1)\), for all \(x, y > 0\).

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