\[ \Large \begin{array} {c c c c }
& & \color{purple}{X}& \color{purple}{X} \\
& & \color{red}{Y} & \color{red}{Y} \\

+ & & \color{blue}{Z} & \color{blue}{Z} \\

\hline
& \color{purple}{X} & \color{red}{Y} & \color{blue}{Z} \\
\end{array} \]

If each letter represents a distinct digit, what is the value of the three-digit number \( \overline{XYZ}? \)

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