Radius of a circle

Geometry Level 5

Let \(ABCD\) be a line with the four points \(A, B, C, D\) in that order such that \(AB=BC=CD=4\). Let \(E, F, G, H\) be points such that \(ABFE, BCGF, CDHG\) are rectangles. Then let \(J, K\) be points such that \(GFJK\) is a rectangle and \(FJ=FB=x\).

If \(x\) is a positive integer, and the radius of the circle passing through \(A, D, J, K\) is the square root of a non-square positive integer, find \(r+S\) where \(\sqrt {r}\) is the largest possible radius of the circle and \(S\) is the sum of all possible values of \(x\).

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