Mathematics Question from JEE Main 2023 · 29 January, Shift 2

Let $\alpha = 8 - 14i$, $A = \left\{ z \in \mathbb{C} : \dfrac{\alpha z - \overline{\alpha}\,\overline{z}}{z^2 - (\overline{z})^2 - 112i} = 1 \right\}$ and $B = \{ z \in \mathbb{C} : |z + 3i| = 4 \}$. Then $\sum_{z \in A \cap B} (\operatorname{Re} z - \operatorname{Im} z)$ is equal to: