
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. Determine all functions \(f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) such that, for all positive real numbers \(x\) and \(y\),
$$
f(x+f(x y))+y=f(x) f(y)+1 .
$$

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