Reflect on this...

Geometry Level 4

A square ABCDABCD of side length kk contains unit circles at each of corners BB and DD such that each circle is tangent to the square at precisely two points. A ray of light emanating from point AA reflects off each circle and then returns to AA, creating a path in the shape of an equilateral triangle.

There is a unique value of kk for which this scenario can occur. Find 10000k\lfloor 10000\cdot k \rfloor.

Note: "Reflecting" means that the angle of incidence equals the angle of reflection.

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