Differential Equations: JEE Main 2023 Mathematics question

Let the solution curve $x=x(y), 0 < y < \frac{\pi}{2}$, of the differential equation $\left(\log _{e}(\cos y)\right)^{2} \cos y \mathrm{~d} x-\left(1+3 x \log _{e}(\cos y)\right) \sin \mathrm{y} d y=0$ satisfy $x\left(\frac{\pi}{3}\right)=\frac{1}{2 \log _{e} 2}$. If $x\left(\frac{\pi}{6}\right)=\frac{1}{\log _{e} m-\log _{e} n}$, where $m$ and $n$ are coprime, then $m n$ is equal to __________.