In how many rearrangements of the numbers $1, \ 2, \ 3, \ 4, \ 5,\  6, \ 7, \ 8,\  9$ do the numbers form a $\textit{hill}$, that is, the numbers form an increasing sequence at the beginning up to a peak, and then form a decreasing sequence to the end such as in $129876543$ or $258976431$?

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