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 ________.

$\frac{1}{\sin \frac{\mathrm{\pi }}{\mathrm{n}}}-\frac{1}{\sin \frac{3\mathrm{\pi }}{\mathrm{n}}}=\frac{1}{\sin \frac{2\mathrm{\pi }}{\mathrm{n}}}$

or   $\frac{\sin \frac{3\mathrm{\pi }}{\mathrm{n}}-\sin \frac{\mathrm{\pi }}{\mathrm{n}}}{\sin \frac{\mathrm{\pi }}{\mathrm{n}}\sin \frac{3\mathrm{\pi }}{\mathrm{n}}}=\frac{1}{\sin \frac{2\mathrm{\pi }}{\mathrm{n}}}$

or   $\frac{\left(2\sin \frac{\mathrm{\pi }}{\mathrm{n}}\cos \frac{2\mathrm{\pi }}{\mathrm{n}}\right)\sin \frac{2\mathrm{\pi }}{\mathrm{n}}}{\sin \frac{\mathrm{\pi }}{\mathrm{n}}\sin \frac{3\mathrm{\pi }}{\mathrm{n}}}=1$

or   $\sin \frac{4\mathrm{\pi }}{\mathrm{n}}=\sin \frac{3\mathrm{\pi }}{\mathrm{n}}$

$\begin{array}{l}\Rightarrow \frac{4\mathrm{\pi }}{\mathrm{n}}+\frac{3\mathrm{\pi }}{\mathrm{n}}=\mathrm{\pi }\\ \Rightarrow \mathrm{n}=7.\end{array}$