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 $m/n$ , 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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