Saketh wants to bring a cake, shaped like a rectangular prism, to Nebraska in order to celebrate the awesomeness of Mitchell. However, he must do this by taking the cake through a secret path, which consists of two cylindrical tunnels joined at an angle of $120$ degrees (in a V shape). Both cylindrical tunnels have a radius of $8$. Assume that the length of each tunnel is significantly greater than the radii and that Saketh takes up no space. Find the volume of the largest cake that Saketh can bring to Mitchell.