2019 AIME II Problem 10

There is a unique angle \theta between 0^\circ and 90^\circ such that for nonnegative integers n, the value of \tan(2^n\theta) is positive when n is a multiple of 3, and negative otherwise. The degree measure of \theta is \tfrac{p}{q}, where p and q are relatively prime positive integers. Find p+q.

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.