The numbers from $1$ to $12$ can be arranged into a $3\times 4$ matrix in either row-major or column-major order: $$R=\left(\begin{array}{cccc}1& 2& 3& 4\\ 5& 6& 7& 8\\ 9& 10& 11& 12\end{array}\right),C=\left(\begin{array}{cccc}1& 4& 7& 10\\ 2& 5& 8& 11\\ 3& 6& 9& 12\end{array}\right)$$ By swapping two entries at a time, at least $8$ swaps are needed to transform $R$ to $C$.

Let $S(n,m)$ be the minimal number of swaps needed to transform an $n\times m$ matrix of $1$ to $nm$ from row-major order to column-major order. Thus $S(3,4)=8$.

You are given that the sum of $S(n,m)$ for $2\le n\le m\le 100$ is $12578833$.

Find the sum of $S(n^4,m^4)$ for $2\le n\le m\le 100$.