
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 \(\mathbb{R}\) denote the set of all real numbers and \(\mathbb{R}^{+}\) the subset of all positive real numbers. Let \(\alpha\) and \(\beta\) be given elements in \(\mathbb{R}\), not necessarily distinct. Find all functions \(f: \mathbb{R}^{+} \rightarrow \mathbb{R}\) such that
\[
f(x) f(y) = y^{\alpha} f\left(\frac{x}{2}\right) + x^{\beta} f\left(\frac{y}{2}\right) \quad \text{for all } x \text{ and } y \text{ in } \mathbb{R}^{+}.
\]

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