Consider the hyperbola $$ \frac{x^{+}}{100}-\frac{y^{-}}{64}=1 $$ with foci at S and $\mathrm{S}_1$, where S lies on the positive x -axis. Let P be a point on the hyperbola, in the first quadrant. Let $\angle \mathrm{SPS}_1=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point S and having the same slope as that of the tangent at P to the hyperbola, intersects the straight line $\mathrm{S}_1 \mathrm{P}$ at $\mathrm{P}_1$. Let $\delta$ be the distance of P from the straight line $\mathrm{SP}_1$, and $\beta=\mathrm{S}_1 \mathrm{P}$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is $\_\_\_\_$ .