2021 AMC 12B Problem 23

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 pq, where p and q are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins 3,17, and 10.) What is p+q?

\textbf{(A) }55 \qquad \textbf{(B) }56 \qquad \textbf{(C) }57\qquad \textbf{(D) }58 \qquad \textbf{(E) }59

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.