Reprocessing is another thing entirely. It relies on the fact that used nuclear fuel is mainly not reusable due to accumulation of neutron poisons. In reprocessing, these poisons are removed so that the rest of the fuel can be reused. This too has many practical challenges as you end up with liquid radioactive materials and a lot of it is waste. So far it has proven more practical than breeders, though. Perhaps that is because the same processes can be used to extract materials for nuclear weapons.