Growing digits

Logic Level 1

\[ \begin{array} { rrrrr } & & & & & A \\ & & & & A & A \\ & & & A & A & A \\ + & & A & A & A & A \\ \hline & \large\square & \large\square & \large\square & \large\square & \large\square \\ \end{array} \]

The above is an incomplete cryptogram, where \(A\) is a positive, single-digit integer. As shown, the final sum is a 5-digit number.

What is this final sum?

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