Let z_1, z_2 and z_3 be three complex numbers on the circle | z |=1 with ( z_1)=(-π)/4, ( z_2)=0 and (z_3)=(π)/4. If |z_1 z̄_2+z_2 z̄_3+z_3…

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