
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
Given the probability $P(b)$ that $\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1$, where $x$ and $y$ are chosen independently at random from the intervals $[0, b]$ (mod 1) and $[0, 1]$, respectively, find the maximum value of $P(b)$.

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