Triangle A B C has A C=450 and B C=300. Points K and L are located on \overline{A C} and \overline{A B} respectively so that A K=C K, and \overline{C L} is the angle bisector of angle C. Let P be the point of intersection of \overline{B K} and \overline{C L}, and let M be the point on line B K for which K is the midpoint of \overline{P M}. If A M=180, find L P.