
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 $b$ with $0 \leq b \leq 1$, let $x$ be chosen independently at random from the interval $[0, 2b]$ and $y$ be chosen from the interval $[0, \frac{1}{2}]$. Calculate the probability $Q(b)$ that
\[
\sin^2{(\pi x)} + \cos^2{(\pi y)} > 1
\]
What is the maximum value of $Q(b)?$
A) $\frac{1}{3}$
B) $\frac{1}{2}$
C) $\frac{5}{8}$
D) $\frac{7}{12}$
E) $\frac{3}{4}$

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