+93 Consider a cube with side length 1, together with 14 congruent spheres each of radius r, with 8 of the spheres centered at the 8 vertices of the cube and the remaining $6$ spheres centered at the centers of the $6$ faces of the cube. Suppose $r$ is chosen so that each sphere centered at a vertex is externally tangent to each sphere centered at the center of an adjacent face. FInd the area of the spheres as a fraction of the areas of the cube.
