Given the system of equations below, find \(x+y+z\).

\(\frac{4}{x} + \frac{1}{y} + \frac{1}{z} = 5\)

\(\frac{1}{x} + \frac{1}{y} - \frac{1}{z} = 4\)

\(\frac{6}{x} - \frac{1}{y} - \frac{1}{z} = 9\)

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