Question 1 Let $n$ be a positive integer, $D_{n}$ be the set of all positive divisors of $2^{n} 3^{n} 5^{n}$, $S \subseteq D_{n}$, and any number in $S$ cannot divide another number in $S$. Find the maximum value of $|S|$. ${ }^{[1]}$
The answer to this question is $\left[\frac{3(n+1)^{2}+1}{4}\right]$, where $[x]$ represents the greatest integer not exceeding the real number $x$.

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The answer to this question is $\left[\frac{3(n+1)^{2}+1}{4}\right]$, where $[x]$ represents the greatest integer not exceeding the real number $x$.