
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 \(\mathbb{R}_{>0}\) be the set of positive real numbers. Find all functions \( f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0} \) such that, for every \( x \in \mathbb{R}_{>0} \), there exists a unique \( y \in \mathbb{R}_{>0} \) satisfying
\[ 
x f(y) + y f(x) \leq 2 . 
\]

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