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

$X,Y,$ and $Z$ are distinct single digits that satisfy the cryptogram above.

What is $X+Y+Z?$

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