
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 \( a, b \in \mathbb{R} \). Find all functions \( f: \mathbb{R}^{+} \rightarrow \mathbb{R} \) such that for all positive real numbers \( x \) and \( y \):

\[ 
f(x) f(y) = y^{a} f\left(\frac{x}{2}\right) + x^{b} f\left(\frac{y}{2}\right) 
\]

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