
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
Given that \(x > 0, y > 0\), and satisfying the following system of equations:
\[
\begin{cases}
\cos ^{2}(\pi x) + 2 \sin (\pi y) = 1 \\
\sin (\pi x) + \sin (\pi y) = 0 \\
x^{2} - y^{2} = 12
\end{cases}
\]
find the ordered pair \((x, y)\).

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