
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  $f:R^+ \rightarrow R^+$  such that  $f(f(x) + y)f(x) = f(xy + 1)\ \ \forall x, y \in R^+$ @below: [https://artofproblemsolving.com/community/c6h2254883_2020_imoc_problems](https://artofproblemsolving.com/community/c6h2254883_2020_imoc_problems)

<blockquote>Feel free to start individual threads for the problems as usual</blockquote>

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