You are given a rectangle on the plane with vertices at integer points. Calculate the area of the largest ellipse that can fit entirely within this rectangle. Recall that a \textit{rectangle} is a convex quadrilateral with all right angles, and an \textit{ellipse} is a figure on the plane defined by two focal points $F_1$ and $F_2$ (not necessarily with integer coordinates; possibly $F_1 = F_2$) and a real number $d > |F_1 F_2|$, consisting of points $P$ on the plane such that $|F_1 P| + |F_2 P| \le d$.