Lattice walk back to origin before the square edge

A particle starts at $(0,0)$ on the integer grid and at each step moves one unit up, down, left or right, each with probability $\tfrac14$. Let $S$ be the boundary of the square whose corners are $(\pm2,\pm2)$. What is the probability that the particle comes back to $(0,0)$ before it first touches $S$?

$1/3$. After the first step the particle is at a neighbour like $(1,0)$; let $p$ be the success probability from there and $q$ from a corner like $(1,1)$. Then $q=\tfrac12p$ (two of its moves hit $S$) and $p=\tfrac14\cdot1+\tfrac14\cdot0+\tfrac12q$, so $p=\tfrac14+\tfrac14p$, i.e. $p=\tfrac13$.