1) "Retain the basic shape" might include squaring it up just a bit. Is this specifically proscribed?

2) The total volume is not 3.846. It's more than that, but less than 3.912. Knowing where to start might be the first hurdle to overcome.

3) Thought question: Are you sure you have to work with the whole thing at the same time. Will you get the same answer if you put 3D coordinate axes in the middle and work with the first octant? If that will work, and you need to decide if it will, that might square things up a bit for the arithmetic.