Grid Shuffle

You are given a square grid $G$ with $N$ rows (numbered $1$ through $N$) and $N$ columns (numbered $1$ through $N$). Let's denote the element in the $i$-th row and $j$-th column by $G_{i, j}$. Initially, for each valid $i, j$, the element $G_{i, j}$ is equal to $i \cdot N+j$. The grid is changing in steps. In one step, either one row or one column of the grid is selected uniformly randomly and th