Alfred and Bonnie play a game where they take turns tossing a fair coin. The winner of the game is the first person to obtain a head. Alfred and Bonnie play this game multiple times with the rule that the loser of a game goes first in the next game. Suppose Alfred goes first in the first game, and 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 $m + n$?

Let's think step by step and output the final answer within \boxed{}.
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