Hmm 5, 12, 13, this seems familiar

Algebra Level 5

Find the sum of \( x \) of all possible values of triples of positive real values \( (x,y,z) \) such that:

  1. \( 5( x+ \frac{1}{x}) = 12(y + \frac{1}{y}) = 13(z+ \frac{1}{z}) \)

  2. \(xy+yz+zx = 1 \)

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