Let AB be a diameter of a circle, and let C be a point on the circle such that AC = 8 and BC = 4 The angle bisector of angle ACB intersects the circle at point M. Find CM.
[asy]
unitsize(0.5 cm);
pair A, B, C, D, P;
A = (4,0);
B = (0,3);
C = (0,0);
D = (12/7,12/7);
P = intersectionpoint(D--(D + 3*(D - C)), circumcircle(A,B,C));
draw(A--B--C--cycle);
draw(circumcircle(A,B,C));
draw(C--P);
label("$A$", A, SE);
label("$B$", B, NW);
label("$C$", C, SW);
dot("$M$", P, NE);
[/asy]
