
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
Determine all functions \( f: (0, \infty) \rightarrow (0, \infty) \) (mapping the set of positive real numbers to the set of positive real numbers) such that the following holds:

$$
\frac{(f(w))^{2}+(f(x))^{2}}{f\left(y^{2}\right)+f\left(z^{2}\right)}=\frac{w^{2}+x^{2}}{y^{2}+z^{2}}
$$

whenever \( w, x, y, z \) are positive real numbers satisfying \( w x=y z \).

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