Similar to AMC 10A #25

Calculus Level 5

Let \(S\) be a square of side length \(1\). Two points \(A\) and \(B\) are chosen independantly at random such that \(A\) is on the perimeter while \(B\) is strictly inside the square. The probability that the straight-line distance between \(A\) and \(B\) 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 \(a+b+c\)?

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