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