If the local maximum value of the function $f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right)$ , is $\frac{k}{e}$, then $\left(\frac{k}{e}\right)^{8}+\frac{k^{8}}{e^{5}}+k^{8}$ is equal to