Question 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 $DE \lt AB,$ $AE = 20,$ $BC = 14\sqrt{2},$ and points $B,$ $C,$ and $D$ lie on the same side of line $AE.$ Suppose further that $AB$ is an integer with $AB \lt 2026$ and the area of pentagon $ABCDE$ is an integer multiple of $16.$ Find the number of possible values of $AB.$