A cardboard box in the shape of a rectangular parallelepiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimeters and its base has a radius of $30$ centimeters, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

Let's think step by step and output the final answer within \boxed{}.
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