2023 AIME I Problem 8

Rhombus ABCD has \angle BAD \lt 90^\circ. There is a point P on the incircle of the rhombus such that the distances from P to the lines DA, AB, and BC are 9, 5, and 16, respectively. Find the perimeter of ABCD.

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.