Let \(y = y(x)\) satisfy
\[ \cos x\,(\log(\cos x))^2\,dy + \left( \sin x - 3y\sin x\log(\cos x) \right)dx = 0, \quad x \in \left(0,\frac{\pi}{2}\right). \]
If \[ y\!\left(\frac{\pi}{4}\right) = -\frac{1}{\log 2}, \] then \[ y\!\left(\frac{\pi}{6}\right) \] equals: