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Problem Description Suppose that \(x,\) \(y,\) and \(z\) are three positive numbers that satisfy the equations \(xyz=1,\) \(x+\frac{1}{z}=5,\) and \(y+\frac{1}{x}=29.\) Then \(z+\frac{1}{y}=\frac{m}{n},\) where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n.\)

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Solution 005

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