Let $P$ be the product of all non-zero digits of the positive integer $n$. For example, $P(4) = 4$, $P(50) = 5$, $P(123) = 6$, $P(2009) = 18$.  
Find the value of the sum: $P(1) + P(2) + \ldots + P(2008) + P(2009)$.

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