In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 8$, $AY = 6$, and $CY = 4$, find $BC$. [asy]
pair A, B,C, X,Y, Z,I;

A= (0,0);
B = (1,0);
C = (0.8,0.7);
X = intersectionpoint(B--C , A -- (bisectorpoint(B, A,C)));
Y = intersectionpoint(A--C, B -- scale(6)*( (bisectorpoint(C, B,A)) - B));
I = intersectionpoint(A--X, B--Y);
Z = extension(A, B,C, I);
draw(A--B--C--cycle);
draw(B--Y);
draw(C--Z);
label("$A$",A, SW);
label("$B$",B, SE);
label("$C$",C, N);
label("$I$",I, SW);
label("$Y$",Y, NW);
label("$Z$",Z, S);
[/asy]