
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 \( R \) be the set of all real numbers, and \( R^{+} \) the subset of all positive real numbers. Given real numbers \( \alpha \) and \( \beta \), find all functions \( f: R^{+} \rightarrow R \) such that for all \( x, y \in R^{+} \),
\[ 
f(x) f(y) = y^{\alpha} f\left(\frac{x}{2}\right) + x^{\beta} f\left(\frac{y}{2}\right).
\]

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