The positive integer value of n > 3 satisfying the equation $\frac{1}{\sin \left(\frac{\mathrm{\pi }}{\mathrm{n}}\right)}=\frac{1}{\sin \left(\frac{2\mathrm{\pi }}{\mathrm{n}}\right)}+\frac{1}{\sin \left(\frac{3\mathrm{\pi }}{\mathrm{n}}\right)}$ is _____

$\begin{array}{l}\frac{1}{\sin \frac{\mathrm{\pi }}{\mathrm{n}}}=\frac{1}{\sin \frac{3\mathrm{\pi }}{\mathrm{n}}}+\frac{1}{\sin \frac{2\mathrm{\pi }}{\mathrm{n}}}\\ \frac{\sin {\displaystyle \frac{3\pi }{n}}+\sin {\displaystyle \frac{2\pi }{n}}}{\sin {\displaystyle \frac{2\pi }{n}}\sin {\displaystyle \frac{3\pi }{n}}}=\frac{1}{\sin {\displaystyle \frac{\pi }{n}}}\\ \Rightarrow \sin \frac{3\pi }{n}+\sin \frac{2\pi }{n}=2\sin \frac{3\pi }{n}\cos \frac{\pi }{n}\end{array}$

$$\begin{gathered} \sin \frac{3\pi }{n}+\sin \frac{2\pi }{n}=\sin \frac{4\pi }{n}+\sin \frac{2\pi }{n} \\ \Rightarrow \sin \frac{3\pi }{n}=\sin \frac{4\pi }{n} \\ \frac{3\pi }{n}+\frac{4\pi }{n}=\pi \Rightarrow n=7 \end{gathered}$$