As easy as XYZ

$\Large \begin{array} {c c c c } & & \color{#69047E}{X}& \color{#69047E}{X} \\ & & \color{#D61F06}{Y} & \color{#D61F06}{Y} \\ + & & \color{#3D99F6}{Z} & \color{#3D99F6}{Z} \\ \hline & \color{#69047E}{X} & \color{#D61F06}{Y} & \color{#3D99F6}{Z} \\ \end{array}$

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

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