Question 12
Circle $C_0$ has radius $1,$ and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r \lt 1$ and is internally tangent to $C_0$ at point $A_0.$ Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^\circ$ counterclockwise from $A_0$ on $C_1.$ Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1.$ In this way a sequence of circles $C_1,$ $C_2,$ $C_3,$ $\ldots$ and a sequence of points on the circles $A_1,$ $A_2,$ $A_3,$ $\ldots$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1},$ and point $A_n$ lies on $C_n$ $90^\circ$ counterclockwise from point $A_{n-1},$ as shown in the figure below. There is one point $B$ inside all of these circles. When $r = \frac{11}{60},$ the distance from the center of $C_0$ to $B$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
