[Perpendicular Planes]

The base of the pyramid $M B K H E$ is a convex quadrilateral $M B K H$, where the angle at vertex $M$ is $\pi / 2$, the angle formed by the diagonal $B H$ and the edge $B K$ is $\pi / 4$, the length of the edge $M B$ is 1. The area of triangle $B K H$ is twice the area of triangle $M B H$. The sum of the lengths of edges $B E$ and $H E$ is $\sqrt{3}$. The volume of the pyramid is 1/4. Find the radius of the sphere with the largest volume among all spheres that can fit inside the pyramid $M B K H E$.
See the given problem.

## Answer

$2-\sqrt{2}$.