Let the solution curve \( x=x(y) \), \( 0<y<\dfrac{\pi}{2}\), of the differential equation \((\log_e(\cos y))^2 \cos y \, dx - (1 + 3x \log_e(\cos y)) \sin y \, dy = 0\) satisfy \(x\left(\dfrac{\pi}{3}\right) = \dfrac{1}{2\log_e 2}\).If \(\displaystyle x\left(\dfrac{\pi}{6}\right) = \dfrac{1}{\log_e m - \log_e n}\),where \(m\) and \(n\) are coprime, then \(mn\) is equal to:
