AIME 2026b 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 $AB$. [asy] size(7cm); import geometry; pair A = (0,0); pair E = (20,0); pair B = (0, 17); pair D = (20, 9); pair C = (14, 3); draw(A--B--C--D--E--cycle); draw(A--E, dashed); dot(A); dot(B); dot(C); dot(D); dot(E); label("A", A, SW); label("B", B, W); label("C", C, N); label("D", D, N); label("E", E, SE); markrightangle(E,A,B); markrightangle(D,E,A); markangle(A,B,C, radius=10); markangle(C,D,E, radius=10); label("$20$", midpoint(A--E), S); label("$14\sqrt{2}$", midpoint(B--C), NE);[/asy] ~Diagram by sillybone