Each positive integer $n \geq 1$ is assigned the number $p_n$, which is the product of all its non-zero digits. For example, $p_6 = 6$, $p_{32} = 6$, $p_{203} = 6$. Let $S = p_1 + p_2 + p_3 + \dots + p_{999}$. Find the largest prime that divides $S$.

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