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 $$x$$.

×