Complex Numbers

Let \(S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\}\). Let \(|z - 4i|\) attains minimum and maximum values, respectively, at z₁ \(\in\) S and z₂ \(\in\) S. If \(5(|{z_1}{|^2} + |{z_2}{|^2}) = \alpha + \beta \sqrt 5\), where \(\alpha\) and \(\beta\) are integers, then the value of \(\alpha\) + \(\beta\) is equal to ___________.