If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg\left(\frac{1+z}{1+\bar{z}}\right)$ equals
(1) $-\theta$
(2) $\frac{\pi}{2}-\theta$
(3) $\theta$
(4) $\pi-\theta$

Let $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$ be three complex numbers on the circle $| z | = 1$ with $\arg \left( z _ { 1 } \right) = \frac { - \pi } { 4 } , \arg \left( z _ { 2 } \right) = 0$ and $\arg \left( z _ { 3 } \right) = \frac { \pi } { 4 }$. If $\left| z _ { 1 } \bar { z } _ { 2 } + z _ { 2 } \bar { z } _ { 3 } + z _ { 3 } \bar { z } _ { 1 } \right| ^ { 2 } = \alpha + \beta \sqrt { 2 } , \alpha , \beta \in \mathbf { Z }$, then the value of $\alpha ^ { 2 } + \beta ^ { 2 }$ is:
(1) 24
(2) 29
(3) 41
(4) 31