Let $S = (0, 2\pi) - \left\{ \frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4} \right\}$. Let $y = y(x), x \in S$, be the solution curve of the differential equation
$\frac{dy}{dx} = \frac{1}{1 + \sin 2x}, y\left(\frac{\pi}{4}\right) = \frac{1}{2}.$
If the sum of abscissas of all the points of intersection of the curve $y = y(x)$ with the curve ($y = \sqrt{2} \sin x$) is $\frac{k\pi}{12}$, then $k$ is equal to ___________.
