42nd Balkan Mathematical Olympiad

Let $△ABC$ be an acute-angled triangle with orthocenter $H$ and let $D$ be an arbitrary point on side $BC$. Let $E$ and $Z$ be the points on the segments $AB$ and $AC$ respectively such that the quadrilaterals $ABDZ$ and $ACDE$ are cyclic, and let segments $BZ$ and $CE$ intersect at $P$. Let $L$ be the intersection point of line $HA$ and the tangent to the circumcircle of triangle $△PBC$ at point $C$. If lines $BH$ and $CP$ intersect at $X$, prove that $D$ lies on the line $LX$.