Question:

Consider the hyperbola $\frac{x^2}{100} - \frac{y^2}{64} = 1$ with foci $S$ and $S_1$ (where $S$ is on the positive $x$-axis). Let $P$ be a point on the hyperbola in the first quadrant and $\angle SPS_1 = \alpha$ with $\alpha < \pi/2$. A line through $S$ with the same slope as the tangent to the hyperbola at $P$ meets line $S_1P$ at $P_1$. If $\delta$ is the distance of $P$ from line $SP_1$ and $\beta = S_1P$, then the greatest integer $\leq \frac{\beta\delta}{9} \sin \frac{\alpha}{2}$ is _____.