Let $ABC$ be any triangle. Equilateral triangles $BCX$, $ACY$, and $BAZ$ are constructed such that none of these triangles overlaps triangle $ABC$. The perimeter of hexagon $AZBXCY$ is $78$. What is the perimeter of $\triangle ABC?$ [asy] pair A,B,C,X,Y,Z; A = (0,0); B = (1,0); C = (0.35,0.7); X = B + (rotate(-60)*(C-B)); Y = rotate(60)*C; Z = rotate(-60)*B; draw(A--B--C--A--Y--C--X--B--Z--A); label("$A$",A,SW); label("$Y$",Y,NW); label("$C$",C,N); label("$X$",X,NE); label("$B$",B,SE); label("$Z$",Z,S); [/asy]
