
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
Let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, \pi]$ and $[-\frac{\pi}{2}, \frac{\pi}{2}]$, respectively. Define $P(\alpha)$ as the probability that
\[\cos^2{x} + \cos^2{y} < \alpha\]
where $\alpha$ is a constant with $1 < \alpha \leq 2$. Find the maximum value of $P(\alpha)$.
A) $\frac{\pi}{4}$ 
B) $\frac{3\pi}{4}$
C) $\pi$
D) $\frac{\pi}{2}$
E) 1

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