A Particle Walks to the Origin

A particle starts at $(4, 4)$ and stops upon first touching a coordinate axis. From $(a, b)$ it moves to $(a-1, b)$, $(a, b-1)$, or $(a-1, b-1)$, each with probability $\tfrac13$, independently. The probability that it stops exactly at the origin is $\dfrac{m}{3^n}$ in lowest terms. Find $m + n$.