$\Large \begin{array} {c c c c } & & \color{purple}{X}& \color{purple}{X} \\ & & \color{red}{Y} & \color{red}{Y} \\ + & & \color{blue}{Z} & \color{blue}{Z} \\ \hline & \color{purple}{X} & \color{red}{Y} & \color{blue}{Z} \\ \end{array}$

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

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