Two skaters, Allie and Billie, are at points A and B, respectively, on a flat, frozen lake. The distance between A and B is 100 meters. Allie leaves A and skates at a speed of 8 meters per second along a straight line that makes an angle of 60^{\circ} with \overline{A B}, as shown. At the same time that Allie leaves A, Billie leaves B at a speed of 7 meters per second and follows the straight line path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?
