Forced Particles

You are given a grid with $n$ rows and $m$ columns. Each cell of this grid can either be empty or it contains one particle. It can never contain more than one particle. Let's denote the cell in the $i$-th row and $j$-th column by $(i, j)$, with the top left corner being $(0, 0)$. From a cell $(i, j)$, a particle could move in one of the following four directions: - to the left, i.e. to the cell $