Logic Level 4

Given any positive integer $$n$$, let $$p(n)$$ be the product of the non-zero digits of $$n$$. If $$n$$ has only one digit, then $$p(n)$$ is equal to that digit.

$$Let: S = p(1) + p(2) + p(3) +\ldots + p(999)$$.

What is the largest prime factor of $$S$$?

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