
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:
\[
\begin{cases}
3 y - \sqrt{\frac{y}{x}} - 6 \sqrt{x y} + 2 = 0, \\
x^2 + 81 x^2 y^4 = 2 y^2.
\end{cases}
\]

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