Let $S = \{1, 2, 3, 4, 5, 6\}$. We want to partition $S$ into exactly three non-empty disjoint subsets $A, B, C$ such that the elements $1$ and $2$ are in different subsets, and the elements $3$ and $4$ are also in different subsets. In how many ways can this partition be formed? (Note: The order of the subsets $A, B, C$ does not matter.)