For $n \in \mathbb{N}$, let $P(n)$ denote the product of the digits in $n$ and $S(n)$ denote the sum of the digits in $n$. Consider the set \(A=\{n \in \mathbb{N}: P(n)\) is non-zero, square-free, and \(S(n)\) is a proper divisor of \(P(n)\}\). Find the maximum possible number of digits of the numbers in \(A\).

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