
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 real-valued functions \( f \) defined on the positive reals such that \( f(x) f(y) = y h f(x/2) + x k f(y/2) \) for all \( x, y \), where \( h \) and \( k \) are real numbers.

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