A circle with integer radius $r$ is centered at $(r,r)$. Distinct line segments of length $c_i$ connect points $(0,a_i)$ to $(b_i,0)$ for $1\le i\le 12$ and are tangent to the circle, where $a_i,b_i$, and $c_i$ are all positive integers and $c_1\le c_2\le \cdots \le c_{12}$. What is the ratio $\frac{c_{12}}{c_1}$ for the least possible value of $r$?