\[ \begin{array}{ c c c c c c c c c } & & A & B& C& D\\ & & D & A& B& C\\ & & C &D & A & B\\ + & & & C & D & A \\ \hline & C & C & C & C & B \end{array} \]

If \(A, B, C, D\) are four distinct digits, how many possible values does \(\overline{ABCD}\) have?

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