
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 $0 \leq a \leq 0.5$, let $x$ and $y$ be chosen independently at random from the intervals $[0, 2a]$ and $[0, 2]$ respectively, and let $Q(a)$ be the probability that
\[\cos(\pi x) + \cos(\pi y) > 1.\]
Find the maximum value of $Q(a)$.
A) 0.75
B) 0.85
C) 0.95
D) 1

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