Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is