IGO 2022 E4

Let $AD$ be the internal angle bisector of triangle $ABC$ . The incircles of triangles $ABC$ and $ACD$ touch each other externally. Prove that $\angle ABC > 120^{\circ}$ . (Recall that the incircle of a triangle is a circle inside the triangle that is tangent to its three sides.)

Proposed by Volodymyr Brayman (Ukraine)