2010015606

The rectangular box $ABCD{A}^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ has edges of lengths $|AB|=4\sqrt{3}\mathrm{cm}$ and $|BC|=8\mathrm{cm}$. The point $S$ is the center of the lateral face $AD{D}^{\prime }{A}^{\prime }$ (see the picture) and the length of the line segment $B^{\prime }S$ is $10\mathrm{cm}$. Find the distance between the points $A$ and $A^{\prime }$.