1993 AIME Problem 11

Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is \frac{m}{n}, where m and n are relatively prime positive integers. What are the last three digits of m+n?