Binary Land

Chefland is a city that always changes. It can be viewed as a grid with a fixed width $N$ (with $N$ columns numbered $1$ through $N$) and changing height. The rows of the grid are numbered with positive integers. Each cell in Chefland has a colour, which is either '0' or '1'. From a cell $(x, y)$, it is possible to move to one of the cells $(x+1, y-1)$, $(x+1, y)$ and $(x+1, y+1)$, but only if