\[\large \begin{array} { l l l l l } & &D & O & G\\ +& &C & A & T \\ \hline & 1 & 0 & 0 & 0 \\ \end{array} \]

The above shows a long addition, where each of the symbols represents a distinct single-digit positive integer.

What is the sum \( D + O + G + C + A + T ? \)

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