2026 AIME I 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 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.