A frog is at the center of a circular shaped island with radius $r$. The frog jumps $\frac{1}{2}$ meters at first. After the first jump, it turns right or left at exactly $90^\circ$, and it always jumps one half of its previous jump. After a finite number of jumps, what is the least $r$ that ensures the frog can never fall into the water? The original answer is in \frac{\sqrt{m}}{n} format, please give the value of m + n.

Let's think step by step and output the final answer within \boxed{}.
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