21. As shown in the figure, curve $C_{1}$ is part of an ellipse centered at the origin $O$ with foci $F_{1}, F_{2}$, and curve $C_{2}$ is part of a parabola with vertex at $O$ and focus at $F_{2}(1,0)$. Point $A\left(\frac{3}{2}, \sqrt{6}\right)$ is the intersection of curves $C_{1}$ and $C_{2}$.
(1) Find the equations of the ellipse and the parabola on which curves $C_{1}$ and $C_{2}$ lie;
(2) Draw a line through $F_{2}$ that is not perpendicular to the $x$-axis, intersecting curves $C_{1}$ and $C_{2}$ at points $B, C, D, E$ in sequence.
(i) Find the range of the area of $\triangle C D F_{1}$;
(ii) If $G$ is the midpoint of $C D$ and $H$ is the midpoint of $B E$, is $\frac{|B E| \cdot\left|G F_{2}\right|}{|C D| \cdot\left|H F_{2}\right|}$ a constant? If so, find the constant; if not, explain why.