Vertices of a square $ABCD$ of side $\frac{25}4$ lie on a sphere. Parallel lines passing through points $A,B,C$ and $D$ intersect the sphere at points $A_1,B_1,C_1$ and $D_1$, respectively. Given that $AA_1=2$, $BB_1=10$, $CC_1=6$, determine the length of the segment $DD_1$.