
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 \( x, y, z \in (0,1] \), find the minimum value of the expression

\[ 
A = \frac{(x+2y) \sqrt{x+y-xy} + (y+2z) \sqrt{y+z-yz} + (z+2x) \sqrt{z+x-zx}}{xy + yz + zx} 
\]

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