Let $S$ be the set of all $(\alpha, \beta)$, $\pi < \alpha, \beta < 2\pi$, for which the complex number $\dfrac{1 - i\sin\alpha}{1 + 2i\sin\alpha}$ is purely imaginary and $\dfrac{1 - 2i\cos\beta}{1 + i\cos\beta}$ is purely real. Let $Z = a\sin 2\alpha + ib\cos 2\beta$ for $(a, b) \in S$.

Then $\sum\limits_{(a, b) \in S} (iZ + \bar{Z})$ is equal to: