Let S be the set of lattice points (x, y) \in \mathbb{Z}^{2} such that -10 \leq x, y \leq 10. Let the point (0,0) be O. Let Scotty the Dog’s position be point P, where initially P=(0,1). At every second, consider all pairs of points C, D \in S such that neither C nor D lies on line O P, and the area of quadrilateral O C P D (with the points going clockwise in that order) is 1. Scotty finds the pair C, D maximizing the sum of the y coordinates of C and D, and randomly jumps to one of them, setting that as the new point P. After 50 such moves, Scotty ends up at point (1,1). Find the probability that he never returned to the point (0,1) during these 50 moves.