A high school basketball game between the Lions and Tigers was tied at the end of the first quarter. The number of points scored by the Lions in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Tigers in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Lions had won by three points. Neither team scored more than $120$ points in total. What was the total number of points scored by the two teams in the first half?