$(25$ pts$.)$ A reversible Carnot heat engine absorbs $800$ kJ of heat from a reservoir at $1173$ K and rejects heat to a reservoir at $300$ K $.$ The entire work output from the engine is used to operate a reversible Carnot refrigerator that absorbs heat from a refrigerated space at $268$ K and rejects heat to the reservoir at $300$ K$.$ Calculate:

$($a$)$ The efficiency of the engine, $.$

$($b$)$ The coefficient of performance of the refrigerator, COP.

$($c$)$ The amount of heat rejected by the engine to the reservoir at $300$ K $.$

$($d$)$ The work produced by the engine.

$($e$)$ The amount of heat removed from the refrigerated space.

$($f$)$ The amount of heat rejected by the refrigerator to the reservoir at $300$ K $.$