A spider is in a huge rectangular room oriented north$/$south and east$/$west$.$ It is $2$ inches east and $4$ inches north of the southwest corner. It repeatedly moves as follows: w$.$p$.$ it moves $2$ inches south; with complementary probability it moves $2$ inches north. Next, w$.$p$.$ it moves $2$ inches west; w$.$p$.$ it moves $2$ inches east; with remaining probability it heads north for an inch but then turns around and goes south for an inch. Its $3$rd move is north$/$south with the same probabilities as the $1$st move; its $4$th move is east$/$west like the $2$nd$,$ etc.

The south and east walls of the room are lined with sticky tape. If the spider ever reaches either wall, it is trapped. If not, the spider escapes. Find the probability that the spider escapes. Hint: all moves are independent of previous moves $($except$,$ of course, that the spider stops moving if it gets trapped.$)$