
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 pairs of positive numbers \((x, y)\) that satisfy the system of equations:

\[
\left\{
\begin{array}{l}
x - 3 \sqrt{x y} - 2 \sqrt{\frac{x}{y}} + 6 = 0, \\
x^2 y^2 + x^4 = 82.
\end{array}
\right.


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