Let $\triangle ABC$ have $AB=3$, $AC=5$, and $\angle A=90^\circ$. Point $D$ is the foot of the altitude from $A$ to $\overline{BC}$, and $X$ and $Y$ are the feet of the altitudes from $D$ to $\overline{AB}$ and $\overline{AC}$ respectively. If $XY^2$ can be written in the form $\frac{m}{n}$ where $m$ and $n$ are positive relatively prime integers, what is $100m+n$?

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