Let's call a natural number \textit{interesting} if any of its two consecutive digits form a number that is a multiple of $19$ or $21$. For example, the number $7638$ is interesting because $76$ is a multiple of $19$, $63$ is a multiple of $21$, and $38$ is a multiple of $19$. How many interesting numbers with $2022$ digits exist?

Let's think step by step and output the final answer within \boxed{}.
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