How is cell division possible?

Consider the following engineering problem. You are told to build a machine that can split itself into two copies. If the machine had access to parts outside of itself, I can picture it being able to assemble the parts into a version of itself. But that is not what is happening. It is more like having the machine cannibalize parts of itself to build a scaled down copy while leaving behind a somewhat diminished version of itself. Conceptually, how is such a thing possible?

The only way I can think of it as possible is to imagine that there is enough slop outside of the nucleus of the cell to use as spare parts to construct a second nucleus. Upon division, the two child cells will not have enough excess material to divide again, but will do so after accumulating enough material through growth