Power Set Width

Let \( X \) be a set of subsets of \( \{1,2,3,4\} \) such that no element of \( X \) is completely contained in any other element of \( X \): that is, for any two distinct subsets \( A,B \in X\), \( A \nsubseteq B \) and \( B \nsubseteq A \).

What is the maximum possible value of \( |X| \)?

(Bonus: generalize to \( \{1,2,\ldots,n\} \).)

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