Periods of repeating decimals!

\[{A = 0.\overline{19} + 0.\overline{199}, \quad B = 0.\overline{19} \times 0.\overline{199}}\]

Recall that \(0.\overline{19},\) for example, stands for the repeating decimal \(0.19191919...\) and that the period of a repeating decimal is the number of digits in the repeating part. In this case, the period of \(0.\overline{19}\) is 2.

Find the sum of the periods of \(A\) and \(B\).

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