Consider a Carnot refrigeration cycle executed in a closed system in the saturated liquid$-$vapor mixture region using $0\mathrm{.}96$ kg of refrigerant$-134$a as the working fluid. It is known that the maximum absolute temperature in the cycle is $1\mathrm{.}2$ times the minimum absolute temperature, and the net work input to the cycle is $22$ kJ$.$ If the refrigerant changes from saturated vapor to saturated liquid during the heat rejection process, determine the minimum pressure in the cycle.