Square $ABCD$ has side length $4$. Points $E$ and $F$ are the midpoints of sides $AB$ and $CD$, respectively. Eight $1 \times 2$ rectangles are placed inside the square so that no two of the eight rectangles overlap. If the arrangement of eight rectangles is chosen randomly, then there are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that none of the rectangles crosses the line segment $EF$. Find $m + n$.

Let's think step by step and output the final answer within \boxed{}.
user