FKMO 2014 Problem 2

Let $ABC$ be a isosceles triangle with $AC = BC > AB$ . Let $E, F$ be the midpoints of segments $AC, AB$ , and let $l$ be the perpendicular bisector of $AC$ . Let $l$ meets $AB$ at $K$ , the line through $B$ parallel to $KC$ meets $AC$ at point $L$ , and line $FL$ meets $l$ at $W$ . Let $P$ be a point on segment $BF$ . Let $H$ be the orthocenter of triangle $ACP$ and line $BH$ and $CP$ meet at point $J$ . Line $FJ$ meets $l$ at $M$ . Prove that $AW = PW$ if and only if $B$ lies on the circumcircle of $EFM$ .