1.1.9 ** For a subset $S$ of the set $\{1,2, \cdots, 15\}$, if the positive integer $n$ and $n+|S|$ are both elements of $S$, then $n$ is called a "good number" of $S$. If a set $S$ has an element that is a "good number", then $S$ is called a "good set". Suppose 7 is a "good number" of some "good set" $X$. How many such subsets $X$ are there?
Parse $7+|X| \leqslant 15$, we know $|X| \in\{2,3, \cdots, 8\}$. When $|X|=2$, $7,9 \in X$, and the other 13 numbers do not belong to $X$, at this time there are $\mathrm{C}_{13}^{0}$ sets of $X$ that meet the requirements; when $|X|=3$, apart from 7,10, one of the other 13 numbers belongs to $X$, at this time there are $\mathrm{C}_{13}^{1}$ sets of $X$ that meet the requirements; ...; when $|X|=8$, apart from 7,15, exactly 6 of the other 13 numbers belong to $X$, there are a total of $C_{13}^{6}$ sets of $X$ that meet the requirements. Therefore, there are $\mathrm{C}_{13}^{0}+\mathrm{C}_{13}^{1}+\cdots+\mathrm{C}_{13}^{6}=\frac{1}{2}\left(\mathrm{C}_{13}^{0}+\cdots+\mathrm{C}_{13}^{13}\right)=2^{12}$ sets of $X$ that meet the requirements. The answer to this problem is $2^{12}$.