$XYZ$ is a triangle in which $\angle X$ is obtuse. A point $P$ is taken inside the triangle and $XP$, $YP$, $ZP$ are produced to meet the sides $YZ$, $ZX$, $XY$ at the points $K$, $L$, $M$, respectively. Suppose that $PL = PM$.
Find the angles of triangle $XYZ$, given that
- $XK$, $YL$, $ZM$ are the angle bisectors of triangle $XYZ$
- and that $2\;XK = YL$.