A heat engine operates between two thermal reservoirs:

$$ The high$-$temperature reservoir is at Thot $=500$ K$.$

$$ The low$-$temperature reservoir is at Tcold$=300$ K$.$

During each cycle, the engine absorbs $2000$ J of heat from the hot reservoir and rejects $1200$ J to

the cold reservoir.

$1.$ What is the theoretical maximum $($Carnot$)$ efficiency of the engine?

$2.$ What is the actual efficiency of the engine?

$3.$ How much work does the engine perform in each cycle?

$4.$ Is the engine operating at its maximum theoretical efficiency? Why or why not?