NL-NWO 2019 Problem 3

Points $A, B$ , and $C$ lie on a circle with centre $M$ . The reflection of point $M$ in the line $AB$ lies inside triangle $ABC$ and is the intersection of the angle bisectors of angles $A$ and $B$ . Line $AM$ intersects the circle again in point $D$ . Show that $|CA| \cdot |CD| = |AB| \cdot |AM|$ .