Problem 9.6. In triangle $A B C$, the angles $\angle B=30^{\circ}$ and $\angle A=90^{\circ}$ are known. On side $A C$, point $K$ is marked, and on side $B C$, points $L$ and $M$ are marked such that $K L=K M$ (point $L$ lies on segment $B M$).

Find the length of segment $L M$, if it is known that $A K=4, B L=31, M C=3$.

![](https://cdn.mathpix.com/cropped/2024_05_06_86512448fd32765ac040g-32.jpg?height=327&width=495&top_left_y=1316&top_left_x=479)