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 $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{1+\sin 2 \mathrm{x}}, \mathrm{y}\left(\frac{\pi}{4}\right)=\frac{1}{2}$. If the sum of abscissas of all the points of intersection of the curve $\mathrm{y}=\mathrm{y}(\mathrm{x})$ with the curve $\mathrm{y}=\sqrt{2} \sin \mathrm{x}$ is $\frac{\mathrm{k} \pi}{12}$, then k is equal to $\_\_\_\_$ .