Challenging Analytic Geometry Problem - 3

Two congruent black circles are placed diagonally in a unit square such that they touch each other at the center of the square. We want to place two congruent ellipses (red) along the other diagonal such that they touch the sides of the square and the two black circles as shown in the figure above. The major and minor axes of each of the two ellipses are parallel to the diagonals of the square. Find the maximum possible area $A$ of one of the two red ellipses . As your answer enter $\lfloor 10^4 A \rfloor$ .