2022 AIME I Problem 3

In isosceles trapezoid ABCD, parallel bases \overline{AB} and \overline{CD} have lengths 500 and 650, respectively, and AD=BC=333. The angle bisectors of \angle A and \angle D meet at P, and the angle bisectors of \angle B and \angle C meet at Q. Find PQ.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.