81. The edge of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ has length $a$. Points $M$ and $N$ lie on segments $B D$ and $C C_{1}$, respectively. The line $M N$ forms an angle $\pi / 4$ with the plane $A B C D$ and an angle $\pi / 6$ with the plane $B B_{1} C_{1} C$. Find: a) the length of segment $M N$; b) the radius of the sphere with center on segment $M N$, touching the planes $A B C D$ and $B B_{1} C_{1} C$.
81. a) $a$; б) $\frac{a(2-\sqrt{2})}{2}$.