Reflecting across a Cone (corrected)

A reflective cone of semi-vertical angle of $30^{\circ}$ . Its vertex is at $( 0, 0, 3 )$ , and its axis of symmetry (and direction of opening up) is along the vector $(1,1,- 2 )$ . A light source is placed at point $A(-20,0,0)$ . A light ray radiating from $A$ is to reflect off the surface of the cone, at a point $C (x,y,z)$ , into point $B(18, 36, 10)$ . Find $x + y + z$ . Note that point $C$ is the point on the surface of the cone that minimizes the sum of distances $AC + CB$ .