Washing Windows

There is a building with $N$ floors (numbered $1$ through $N$ from bottom to top); on each floor, there are $M$ windows (numbered $1$ through $M$ from left to right). Let's denote the $j$-th window on the $i$-th floor by $(i, j)$. All windows in the building need to be cleaned one by one in a specific order. You are given this order as a matrix $A$ with $N$ rows and $M$ columns; for each valid