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