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For $A \subset \mathbb{R}$, let $A'=\{a+1|a \in A\}$. Let $S_n=\{1;2;3;...;n\}$. How many subsets $A$ of $S_n$ satisfying $A \cup A' = S_n$?

I'm trying to solve this problem by using bijection or constructing a sequence but still struggling. Please help me with this.

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