Differential Equations

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 __________.