Consider the ellipse$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \qquad (0 < b < a),$and suppose that one focus of this ellipse coincides with the focus of the parabola$y^2 = 4px \qquad (p > 0).$Let the two intersection points of the parabola and the ellipse be denoted by$\mathrm{P}_1$ and $\mathrm{P}_2$.Let $A$ be the intersection point of the tangent line to the parabola at $\mathrm{P}_1$and the tangent line to the parabola at $\mathrm{P}_2$.Given that the angle $\angle \mathrm{P}_1 A \mathrm{P}_2 = 60^\circ$, and that for any real number $t$,$\lfloor t \rfloor$ denotes the greatest integer not exceeding $t$, determine the value of$\left\lfloor \frac{a^3}{p^3} \right\rfloor .$oindent