A positive rational number $x$ is called banzai if the following conditions are met: $\bullet$ $x=\frac{p}{q}>1$ where $p,q$ are comprime natural numbers $\bullet$ exist constants $\alpha,N$ such that for all integers $n\geq N$,$$\mid \left\{\,x^n\right\} -\alpha\mid \leq \dfrac{1}{2(p+q)}.$$Find the total number of banzai numbers. Note:$\left\{\,x\right\}$ means fractional part of $x$ (tatari/nightmare)