2021 AIME I #12

Let A_1A_2A_3...A_{12} be a dodecagon (12-gon). Three frogs sit at A_4, A_8, and A_{12}. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.

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