Three. (50 points) Let $S=\{1,2, \cdots, 100\}$. Find the largest integer $k$, such that the set $S$ has $k$ distinct non-empty subsets with the property: for any two different subsets among these $k$ subsets, if their intersection is non-empty, then the smallest element in their intersection is different from the largest elements in both of these subsets.

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