2026 AIME II Problem 3

Let ABCDE be a nonconvex pentagon with internal angles \angle A = \angle E = 90^\circ and \angle B = \angle D = 45^\circ. Suppose that DE < AB, AE = 20, BC = 14\sqrt2, and points B,C, and D lie on the same side of line AE. Suppose further that AB is an integer with AB < 2026 and the area of pentagon ABCDE is an integer multiple of 16. Find the number of possible values of AB.

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.