In a three$-$stage separation of proteins A and B$,$ it is desired to produce $1$ mg of protein A of $97\%$ purity.

Equal masses $($i$.$e$.$ a $50/50$ mixture$)$ of A and B are added to a separation train of three separators. Stage $1$ is a solvent

extraction with a yield of $83\%$ for A and $17\%$ for protein B$.$ Stage $2$ is gel chromatography with a yield of $62\%$ for A

and $27\%$ for B$.$ Stage $3$ is dialysis with a yield of $77\%$ for A$.$ What is the yield for protein B in stage $3$ to achieve the

desired level of purity? How much protein do you need to add to the beginning of the separator to yield $1$ mg of

protein A$?$