AIME 2026a Problem 5

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 a point $A \prime$. Point $B$ is rotated across a angle of $\theta$ around point $A \prime$ clockwise to a point $B \prime$. $A B \prime = \frac {4}{3}$. If $\cos \theta = \frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.