Counting Tilings - Problem Statement

Given an $n \times m$ grid, count how many ways you can tile it completely using $1 \times 2$ and $2 \times 1$ dominoes. Both dimensions can be up to $10$, and the answer can be huge, so report it modulo $10^9 + 7$.

For example, a $2 \times 3$ grid has $3$ tilings: all horizontal, all vertical, or mixed. You need to count these systematically without missing or double-counting any. This is the canonical broken profile problem. Before reading the solution, think: how would you track which cells are filled as you go column by column?