
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 \( f \) on the positive reals such that \( f(1) = \frac{1}{2} \) and \( f(xy) = f(x) f\left(\frac{3}{y}\right) + f(y) f\left(\frac{3}{x}\right) \).

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