Let A B C D be a square with side length 100 . A circle with center...

Let $A B C D$ be a square with side length 100 . A circle with center $C$ and radius $C D$ is drawn. Another circle of radius $r$, lying inside $A B C D$, is drawn to touch this circle externally and such that the circle also touches $A B$ and $A D$. If $r=m+n \sqrt{k}$, where $m, n$ are integers and $k$ is a prime number, find the value of $\frac{m+n}{k}$.