If \( x, y, z \) are non-zero real numbers and \( \left|\begin{array}{ccc}1+x & 1 & 1 \\ 1+y & 1...

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If \( x, y, z \) are non-zero real numbers and \( \left|\begin{array}{ccc}1+x & 1 & 1 \\ 1+y & 1+2 y & 1 \\ 1+z & 1+z & 1+3 z\end{array}\right|=0 \), then \( \left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \) is equal to
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