Let $y = y(x)$ be the solution of the differential equation $\cos x\left(\log_e(\cos x)\right)^2 \mathrm{d}y + \left(\sin x - 3y\sin x\log_e(\cos x)\right)\mathrm{d}x = 0$, $x \in \left(0, \frac{\pi}{2}\right)$. If $y\left(\frac{\pi}{4}\right) = \frac{-1}{\log_e 2}$, then $y\left(\frac{\pi}{6}\right)$ is equal to:\\
(1) $\frac{1}{\log_e(3) - \log_e(4)}$\\
(2) $\frac{2}{\log_e(3) - \log_e(4)}$\\
(3) $\frac{1}{\log_e(4) - \log_e(3)}$\\
(4) $-\frac{1}{\log_e(4)}$