Random walk exiting a 4 by 4 square sideways

A walker starts at the lattice point $(1,2)$ and performs a simple symmetric random walk: each step goes up, down, left or right with probability $\tfrac14$ each. The walk stops as soon as it reaches the boundary of the square with corners $(0,0)$, $(4,0)$, $(4,4)$ and $(0,4)$. What is the probability that the walk stops on one of the two vertical sides ($x=0$ or $x=4$)?

$\tfrac58$. From any point on a diagonal of the square, reflecting across that diagonal swaps vertical and horizontal sides, so the chance is $\tfrac12$. From $(1,2)$, a step left hits $x=0$ immediately, and each of the other three steps lands on a diagonal, so the answer is $\tfrac14+\tfrac34\cdot\tfrac12=\tfrac58$.