Can You Spot The Pattern?

\[ \large{\begin{array}{cccccccccc} &0.& 1 & 1& 1 & 1&1&1& 1&1&1& \ldots\\ \times&0. & 1 & 1& 1 & 1&1&1&1& 1&1&\ldots\\ \hline &0. &0 &1 &2& 3& 4&5&6&7&\diamondsuit&\ldots\\ \hline \end{array}} \]

The above shows the product of the two fractions \( \dfrac19 \) and \( \dfrac19 \) when written in decimal representation.

Find the value that represents the symbol \( \diamondsuit\).

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