Let $S$ be a square of side length 1. 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 the value of $a+b+c$?

Let's think step by step and output the final answer within \boxed{}.
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