jmc

Triangle $ABC$ has a right angle at $C,AC=3$ and $BC=4$. Triangle $ABD$ has a right angle at $A$ and $AD=12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If$$\frac{DE}{DB}=\frac{m}{n},$$where $m$ and $n$ are relatively prime positive integers, then $m+n=$ $\text{(A) }25\text{(B) }128\text{(C) }153\text{(D) }243\text{(E) }256$