
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 functions \( f: (0, +\infty) \rightarrow (0, +\infty) \) that satisfy, for all positive real numbers \( \omega, x, y, z \) such that \( \omega x = yz \), the equation:
\[ \frac{(f(\omega))^2 + (f(x))^2}{f(y^2) + f(z^2)} = \frac{\omega^2 + x^2}{y^2 + z^2}. \]

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