USAMO 1993 Problem 4

The sequence $a_{n}$ of odd positive integers is defined as follows: $a_{1}=r, a_{2}=s$, and $a_{n}$ is the greatest odd divisor of $a_{n-1}+a_{n-2}$. Show that, for sufficiently large $n, a_{n}$ is constant and find this constant (in terms of $r$ and $s$ ).