smc

If ${\log }_{x}n=0,$ then $n=1.$ So, we can rewrite this equation: $$({\log }_3({\log }_{4}x))=({\log }_4({\log }_{2}y))=({\log }_2({\log }_{3}z))=1$$ Solve individually for each variable. $$x=4^3=64$$ $$y=2^4=16$$ $$z=3^2=9$$ Therefore, $x+y+z=89.$ The answer is $\boxed{89}.$