
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
Given \( x, y, z \) are 3 real numbers greater than or equal to 1, find the sum of the numerator and denominator of the simplest fraction representing the minimum value of

$$\left(\frac{\sqrt{x^{2}(y-1)^{2}+y^{2}}}{x y}+\frac{\sqrt{y^{2}(z-1)^{2}+z^{2}}}{y z}+\frac{\sqrt{z^{2}(x-1)^{2}+x^{2}}}{z x}\right)^{2}$$.

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