A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\theta$ around point $B$ to point $A'$. Then $B$ is rotated in the plane clockwise through angle $\theta$ around point $A'$ to point $B'$. Suppose that $AB' = \frac{4}{3}$. The value of $\cos\theta$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.