A finite sequence of positive integers $m_i$ for $i = 1, 2, \ldots, 2006$ is defined such that $m_1 = 1$ and $m_i = 10m_{i-1} + 1$ for $i > 1$. How many of these integers are divisible by $37$?

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