
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

For \( x, y \in [1,3] \), find the minimum value of the expression

\[ 
A = \frac{(3xy + x^{2}) \sqrt{3xy + x - 3y} + (3xy + y^{2}) \sqrt{3xy + y - 3x}}{x^{2} y + y^{2} x} 
\]

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