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 $\left|\,z_1\overline{z_2}+z_2\overline{z_3}+z_3\overline{z_1}\,\right|^2=\alpha+\beta\sqrt{2}$, $\alpha,\beta\in\mathbb{Z}$, then the value of $\alpha^2+\beta^2$ is: