As $\sin x$ is a convex function for $x \in (0, \frac{\pi}{6})$, we have $\frac{3}{\pi}x < \sin x < x$. Then
$$\sin \frac{\pi}{13} < \frac{\pi}{13} < \frac{1}{4}, \sin \frac{\pi}{12} > \frac{3}{\pi} \times \frac{\pi}{12} = \frac{1}{4},$$
$$\sin \frac{\pi}{10} < \frac{\pi}{10} < \frac{1}{3}, \sin \frac{\pi}{9} > \frac{3}{\pi} \times \frac{\pi}{9} = \frac{1}{3},$$
that is
$$\sin \frac{\pi}{13} < \frac{1}{4} < \sin \frac{\pi}{12} < \sin \frac{\pi}{11} < \sin \frac{\pi}{10} < \frac{1}{3} < \sin \frac{\pi}{9}.$$
Therefore, all the possible values of positive integers $n$ are
$10, 11, 12$, and their sum is 33.
The answer is 33.