CAPS 2013 Problem 6

Let ${ABC}$ be a triangle inscribed in a circle. Point ${P}$ is the center of the arc ${BAC}$ . The circle with the diameter ${CP}$ intersects the angle bisector of angle ${\angle BAC}$ at points ${K, L}$ ${(|AK| <|AL|)}$ . Point ${M}$ is the reflection of ${L}$ with respect to line ${BC}$ . Prove that the circumcircle of the triangle ${BKM}$ passes through the center of the segment ${BC}$ .