Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$ then the average value (arithmetic mean) of the integers remaining is $40$. If the least integer in $S$ is also removed then the average value of the integers remaining is $42$. If the greatest integer is then returned to the set the average value of the integers rises to $50$. The greatest integer in the original set $S$ is $110$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?