A small Carnot engine operates between two identical bodies, each having a finite heat capacity $C_P,$ initially at temperatures $T_1$ and $T_2$ respectively, as shown in the figure below. Heat flows from the higher temperature body to the lower temperature body until the two bodies eventually reach the same temperature, $T_f.$ Calculate the total amount of work done by the Carnot engine before the temperatures of both bodies eventually reach the same temperature $T_f.$ How would the result differ if the cold reservoir was large enough such that its temperature remains constant?

$($Please explain each and every step properly$)$