Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn, the probability that a player knocks the bottle off the ledge is \(\tfrac{1}{2}\), independently of what has happened before. What is the probability that Larry wins the game? Express your answer in the form \(\frac{k}{m}\), and find the value of \(k + m\).

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