
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 \((x, y)\) that satisfy the system of equations 
\[
\left\{
\begin{array}{l}
\frac{\cos x}{\cos y} = 2 \cos^2 y \\
\frac{\sin x}{\sin y} = 2 \sin^2 y 
\end{array}
\right.
\]
where \(x\) and \(y\) are in the interval \(\left(0, \frac{\pi}{2}\right)\).

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