HMMT 2024 G8

Let $A B T C D$ be a convex pentagon with area 22 such that $A B=C D$ and the circumcircles of triangles $T A B$ and $T C D$ are internally tangent. Given that $\angle A T D=90^{\circ}, \angle B T C=120^{\circ}, B T=4$, and $C T=5$, compute the area of triangle $T A D$.