Adding another reciprocal

Algebra Level 4

Given that xyz=1xyz=1 where x,y,zx,y,z are positive real numbers and that x+1z=5x+\dfrac{1}{z}=5, y+1x=29y+\dfrac{1}{x}=29 and z+1y=mnz+\dfrac{1}{y}=\dfrac{m}{n}, where mm and nn are coprime positive integers. Compute m+nm+n.

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