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