Cornered Again! inspired by "We've got you cornered"

Geometry Level 4

Suppose that two circles of equal radius are drawn within a square such that each is centered on the same diagonal of the square and each is tangent to precisely two sides of the square, as well as to one another at precisely one point.

Next, on each end of the other diagonal, circles are drawn such they are both tangent at precisely one point to each of the two previously drawn circles and tangent to two sides of the square.

Let \(S\) be the ratio of the combined areas of the four circles to the area of the square. Find \(S\) to the nearest 3 decimal points.

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