2019 AIME I Problem 5

A moving particle starts at the point (4,4) and moves until it hits one of the coordinate axes for the first time. When the particle is at the point (a,b), it moves at random to one of the points (a-1,b), (a,b-1), or (a-1,b-1), each with probability \tfrac{1}{3}, independently of its previous moves. The probability that it will hit the coordinate axes at (0,0) is \tfrac{m}{3^n}, where m and n are positive integers, and m is not divisible by 3. Find m + n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

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