Let $S$ be a square of side length $2$. Two points are chosen independently at random on the sides of $S$. 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 and $\gcd (a,b,c)=1$. What is $a+b+c$?