Starting from a positive integer $n$, at each step we subtract from $n$ the largest perfect cube not exceeding $n$, until $n$ becomes $0$.
For example, with $n = 100$ the procedure ends in $4$ steps:
$$100 \xrightarrow{-4^3} 36 \xrightarrow{-3^3} 9 \xrightarrow{-2^3} 1 \xrightarrow{-1^3} 0.$$
Let $D(n)$ denote the number of steps of the procedure. Thus $D(100) = 4$.
Let $S(N)$ denote the sum of $D(n)$ for all positive integers $n$ strictly less than $N$.
For example, $S(100) = 512$.
Find $S(3^N)$.