BG-MO 2021 Problem 6

Point $S$ is the midpoint of arc $ACB$ of the circumscribed circle $k$ around triangle $ABC$ with $AC>BC$ . Let $I$ be the incenter of triangle $ABC$ . Line $SI$ intersects $k$ again at point $T$ . Let $D$ be the reflection of $I$ across $T$ and $M$ be the midpoint of side $AB$ . Line $IM$ intersects the line through $D$ , parallel to $AB$ , at point $E$ . Prove that $AE=BD$ .