\[ \begin{array} { l l l l l } & & X & Y& Z\\ & & X& Y& Z\\+&& &Y & Z \\\hline & 1 & 6 & 7 & 5\\ \end{array} \]

\(X,Y,\) and \(Z\) are distinct single digits that satisfy the cryptogram above.

What is \(X+Y+Z?\)

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