Let $n$ be an integer greater than 1, and let $X$ be a set of $n$ elements. If the union $A_{1} \cup \cdots \cup A_{k}$ is a proper subset of $X$, and no element of $X$ is in exactly one of $A_{i}$, then the set of non-empty subsets $A_{1}, \ldots, A_{k}$ of $X$ is compact. Find the maximum cardinality of the set of proper non-empty subsets of $X$ such that no non-empty subset is compact, and compute its value for $n=100$.

Let's think step by step and output the final answer within \boxed{}.
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