Let $A B C D$ be an isosceles trapezoid with $\overline{B C} || \overline{A D}$ and $A B=C D$. Points $X$ and $Y$ lie on diagonal $\overline{A C}$ with $X$ between $A$ and $Y$, as shown in the figure. Suppose $\angle A X D=\angle B Y C=90^{\circ}, A X=3, X Y=1$, and $Y C=2$. What is the area of $A B C D$ ?