
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
Solve the system of equations:
\[
\begin{cases}
\sqrt{\frac{y}{x}} - 2 \sqrt{\frac{x}{y}} = 1 \\
\sqrt{5x + y} + \sqrt{5x - y} = 4
\end{cases}
\]

Solve the system of equations:
\[
\begin{cases}
\sqrt[3]{x} + \sqrt[3]{y} = 3 \\
\sqrt[3]{x^2} - \sqrt[3]{xy} + \sqrt[3]{y^2} = 3
\end{cases}
\]

Solve the system of equations:
\[
\begin{cases}
\sqrt[3]{x} \sqrt{y} + \sqrt[3]{y} \sqrt{x} = 12 \\
xy = 64
\end{cases}
\]

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