
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 all sets of positive integers \((x, y, u, v)\) such that \(\sqrt{x y}+\sqrt{u v}\) is a rational number, and \(\left|\frac{x}{9}-\frac{y}{4}\right|=\left|\frac{u}{3}-\frac{v}{12}\right|=u v-x y\).

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