
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 \( x > 0 \), \( y > 0 \), and the following system of equations:
\[
\left\{
\begin{array}{l}
\cos^2(\pi x) + 2 \sin(\pi y) = 1, \\
\sin(\pi x) + \sin(\pi y) = 0, \\
x^2 - y^2 = 12
\end{array}
\right.
\]

Find the ordered pair \((x, y) = \qquad\).

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