OMCC 2020 Problem 4

Consider a triangle $ABC$ with $BC>AC$ . The circle with center $C$ and radius $AC$ intersects the segment $BC$ in $D$ . Let $I$ be the incenter of triangle $ABC$ and $\Gamma$ be the circle that passes through $I$ and is tangent to the line $CA$ at $A$ . The line $AB$ and $\Gamma$ intersect at a point $F$ with $F \neq A$ . Prove that $BF=BD$ .