Check if this point lies within the circular region:
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- Set $\sin(2\theta) = 1 \Rightarrow 2\theta = \frac{\pi}{2} + 2k\pi \Rightarrow \theta = \frac{\pi}{4} + k\pi$.
- Try $\theta = \frac{\pi}{4}$:
- Then $r^2 = \frac{100}{1} = 100 \Rightarrow r = 10$.
- The circle has radius 10, so $r = 10$ is on the boundary — valid.
- Thus, the **maximum possible value of $r$** is:
- \boxed{10}
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