A certain heat engine operates between $1000$ K and $450$ K$.($a$)$ What is the maximum efficiency of the engine? $($b$)$ Calculate the maximum work that can be done by for each $1\mathrm{.}0$ kJ of heat supplied by the hot source. $($c$)$ How much heat is discharged into the cold sink in a reversible process for each $1\mathrm{.}0$ kJ supplied by the hot source?