
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 \( f: (0,+\infty) \rightarrow \mathbf{R} \) such that for all \( x, y > 0 \), the following holds:
\[ 
\left(x+\frac{1}{x}\right) f(y) = f(x y) + f\left(\frac{y}{x}\right). 
\]

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