
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 the set of values taken by the function \( f:] 0,+\infty[^{3} \mapsto \mathbb{R} \) such that
\[ f(x, y, z) = \frac{x \sqrt{y} + y \sqrt{z} + z \sqrt{x}}{\sqrt{(x+y)(y+z)(z+x)}}. \]

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