A heat engine operates between two reservoirs at $800$C and $20$C$.$ One$-$half of the work output of the heat engine is used to drive a Carnot heat pump that removes heat from the cold surroundings at $2$C and transfers it to a house maintained at $22$C$.$ If the house is losing heat at a rate of $62900$ kJ$/$h$,$ determine the minimum rate of heat supply to the heat engine required to keep the house at $22$C$.$

$$

The minimum rate of heat supply to the heat engine is kJ$/$h$.$