
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 $0 \leq b \leq \frac{1}{2}$, let two numbers $x$ and $y$ be chosen independently at random from the intervals $[0, b]$ and $[0, 2b]$, respectively. Define $Q(b)$ as the probability that
\[\sin(\pi x) \cdot \sin(\pi y) > \frac{1}{2}.\]
What is the maximum value of $Q(b)?$
A) $\frac{1}{8}$
B) $\frac{1}{16}$
C) $\frac{1}{4}$
D) $\frac{1}{32}$
E) $\frac{1}{64}$

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