# The Other AM-GM (Part 1)

Algebra Level 4

$\large (x+ 2y)(y+2z)(xz+1)$

Positive reals $$x$$, $$y$$, and $$z$$ are such that $$xyz = 1$$. If the value of the expression above is minimum at $$(x_m, y_m, z_m)$$ and $$x_m+y_m+z_m = \dfrac mn$$, where $$m$$ and $$n$$ are coprime positive integers. Find $$m+n$$.