TPCW3: Uh-Oh.

Gwen Stacy and Spiderman are separated on a $3$ by $3$ grid at lattice points inside $(0, 0)$ , $(3, 3)$ , $(0,3)$ , and $(3,0)$ . Spiderman is at $(3, 3)$ , and Gwen is at $(0, 0)$ . Spiderman must get to Gwen. Unfortunately, Electro is at $(1,1)$ , and Spiderman can either avoid Electro, or if he gets to Electro, he has a $\frac{2}{3}$ chance of passing easily and safely, and $\frac{1}{9}$ chance of barely passing, but if this happens, he gets thrown back to $(2, 2)$ . Gwen cannot move at all, and Spiderman can only move left or down $1$ unit at a time (with equal probability).

The probability that Spiderman gets to Gwen is $\frac{a}{b}$ where $a$ and $b$ are coprime and positive integers. Find $a+b$ .