(7) The sum of all positive integers $n$ that satisfy $\frac{1}{4}<\sin \frac{\pi}{n}<\frac{1}{3}$ is $\qquad$
(7) 33 Hint: By the convexity of the sine function, when $x \in\left(0, \frac{\pi}{6}\right)$, $\frac{3}{\pi} x$ $\sin x\frac{3}{\pi} \times \frac{\pi}{12}=\frac{1}{4}, \\
\sin \frac{\pi}{10}\frac{3}{\pi} \times \frac{\pi}{9}=\frac{1}{3} .
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Therefore, $\sin \frac{\pi}{13}<\frac{1}{4}<\sin \frac{\pi}{12}<\sin \frac{\pi}{11}<\sin \frac{\pi}{10}<\frac{1}{3}<\sin \frac{\pi}{9}$. Hence, the positive integer values of $n$ that satisfy $\frac{1}{4}<\sin \frac{\pi}{n}<\frac{1}{3}$ are 10, 11, and 12, and their sum is 33.