14. Square $A B C D$ has a side length of $3, E$ is on side $A B$, and $A E=1$. Jia and Yi play the following game: Jia first selects a point $X$ on $B E$, then Yi selects a point $Y$ on side $B C$, and then fold the square along $D X, D Y$ so that points $A, C$ coincide at point $A^{\prime}$. Jia's goal is to make the angle between $A^{\prime} D$ and the plane $D X Y$ as small as possible, while Yi's goal is to make this angle as large as possible. If both players adopt the correct strategy, find the size of the angle between $A^{\prime} D$ and the plane $D X Y$, and the positions of points $X$ and $Y$.
14. When $X$ is determined, if plane $A^{\prime} D X$ is perpendicular to plane $D X Y$, the angle formed by $A^{\prime} D$ and plane $D X Y$ is the largest (by the minimum angle theorem), which equals $\angle A D X$. To minimize this angle, $X$ must coincide with point $E$. Since plane $A D E \perp$ plane $D E Y$, it is easy to prove that $E Y \perp D E$, thus we get $B Y: B E = A E: A D$, which leads to $B Y = \frac{1}{3} B E = \frac{2}{3} \cdot A D$. The angle formed by $A D E$ with plane $D X Y$ is $\angle A D E = \arctan \frac{1}{3}$.