The set $A=\{1,2,3,\cdots, 10\}$ contains the numbers $1$ through $10$. A subset of $A$ of size $n$ is competent if it contains $n$ as an element. A subset of $A$ is minimally competent if it itself is competent, but none of its proper subsets are. Find the total number of minimally competent subsets of $A$.

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