2021 AIME I Problem 12

Let A_{1} A_{2} A_{3} \ldots A_{12} be a dodecagon (12-gon). Three frogs initially 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.