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 \(\tfrac{p}{q}\), where \(p\) and \(q\) are relatively prime integers. Find \(p+q\).