7. For square $A B C D$ with side length $4, E$ is the midpoint of $A B$, and $F$ is a moving point on side $B C$. If $\triangle A D E$ and $\triangle D C F$ are folded along $D E$ and $D F$ respectively, such that $A, C$ coincide at $A^{\prime}$, then the maximum distance from point $A^{\prime}$ to plane $D E F$ is
7. $\frac{4 \sqrt{5}}{5}$  Let $A^{\prime} G \perp D E$ at $G, A^{\prime} H \perp$ plane $D E F$ at $H$, then we have $A^{\prime} H \leqslant A^{\prime} G=\frac{A^{\prime} D \cdot A^{\prime} E}{D E}=\frac{4 \sqrt{5}}{5}$.