INMO 2005 Problem 1

Let $M$ be the midpoint of side $B C$ of a triangle $A B C$. Let the median $A M$ intersect the incircle of $A B C$ at $K$ and $L, K$ being nearer to $A$ than $L$. If $A K=K L=L M$, prove that the sides of triangle $A B C$ are in the ratio $5: 10: 13$ in some order.