Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,....$ More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q.$

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