Let \( z_1, z_2 \) and \( z_3 \) be three complex numbers on the circle \( |z| = 1 \) with \( \arg(z_1) = -\pi/4 \), \( \arg(z_2) = 0 \) and \( \arg(z_3) = \pi/4 \).
If \( \left| z_1\overline{z_2} + z_2\overline{z_3} + z_3\overline{z_1} \right|^{2} = \alpha + \beta \sqrt{2}, \alpha, \beta \in \mathbb{Z} \), then the value of \( \alpha^{2} + \beta^{2} \) is:
