Let $R$ be a rectangle of side lengths $1$ and $2$. Two points are chosen at random on the sides of $R$. The probability that the straight-line distance between the points is at least $\frac{1}{2}$ is $\frac{a-b\pi }{c}$, where $a$, $b$, and $c$ are positive integers with $\gcd (a,b,c)=1$. What is $a+b+c$?