7. Given that the base of the prism $A B C-A^{\prime} B^{\prime} C^{\prime \prime}$ is an equilateral triangle with side length 2, the side edge $A A^{\prime}$ forms a 45-degree angle with the sides $A B, A C$ of the lower base, and $A^{\prime}$ is equidistant from the plane $A B C$ and the plane $B B^{\prime} C^{\prime} C$, then $A^{\prime} A=$ $\qquad$
7. $\sqrt{3}$

Li Nan $V_{A^{\prime}-A B C}=\frac{1}{2} V_{A^{\prime}-B C C^{\prime}}$, we get $S_{\triangle A S C}=\frac{1}{2} S_{F C C^{\prime} B^{\prime}}$, from the given information, it is easy to see that $B C C^{\prime}$ $B^{\prime}$ is a rectangle.