Problem 6. (30 points) A regular triangular prism $A B C A_{1} B_{1} C_{1}$ is inscribed in a sphere with the base $A B C$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$. The segment $C_{1} D$ is the diameter of this sphere, and point $K$ is the midpoint of edge $C C_{1}$. Find the volume of the prism if $D K=2, D A=\sqrt{6}$.