Given a 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

$S=p(1)+p(2)+p(3)+\cdots+p(999)$
.

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



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