
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
For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a^2]$ and $[0, 1]$, respectively, and let $Q(a)$ be the probability that
\[\cos^2{(\pi x)} + \cos^2{(\pi y)} < a.\]
What is the maximum value of $Q(a)?$
A) $\frac{1}{2}$
B) $\frac{\sqrt{2}}{2}$
C) $\frac{3}{4}$
D) $1$

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