Define $p(n)$ to be the product of all non-zero digits of $n$. For instance, $p(5) = 5$, $p(27) = 14$, $p(101) = 1$, and so on. Find the greatest prime divisor of the following expression:
\[
p(1) + p(2) + p(3) + \ldots + p(999).
\]

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