Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $A = \{1, 2, 3, 4\}$. Find the number of $4$-element subsets $Y$ of $X$ such that $10 \in Y$ and the intersection of $Y$ and $A$ is not empty.

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