Lattice walk escaping a square of half-width two

A particle starts at $(0,0)$ and moves on the integer lattice, each step going one unit north, south, east or west with probability $\tfrac14$ each. How many steps does it take on average to first reach the boundary of the square with corners $(\pm2,\pm2)$, i.e. a point with $|x|=2$ or $|y|=2$?

$\tfrac92$. By symmetry only three interior states matter: the centre, an edge-adjacent point like $(1,0)$, and a corner like $(1,1)$. First-step analysis gives $e_0=1+e_1$, $e_1=1+\tfrac14e_0+\tfrac12e_2$, $e_2=1+\tfrac12e_1$, whose solution is $e_1=\tfrac72$, $e_2=\tfrac{11}{4}$ and $e_0=\tfrac92$.