Seeking the triples

\[\frac{1}{x} + \frac{2}{y} - \frac{4}{z} = 1 \]

Find the sum \(S\) of all triples \((x, y, z)\) for \(x, y, z\) positive integers, \(x \neq 1\) and \(y \neq 2\) such that they satisfy the above expression.

Clarification:

If there are 3 triples \((x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3)\) that satisfy the expression, then

\(S = x_1 + y_1 + z_1 + x_2 + y_2 + z_2 + x_3 + y_3 + z_3\)

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