Snakes on a Grid

There is an $N\times N$ grid, with rows numbered $1$ to $N$ from top to bottom and columns numbered $1$ to $N$ from left to right. The cell at the intersection of row $i$ and column $j$ is denoted $(i, j)$. Chef is standing at cell $(1, 1)$ of the grid, and would like to get to cell $(N, N)$. To do this, he can move one step up/down/left/right from the cell he's standing at, as long as he remains