Simplify the fraction

$$\zeta=\frac{1}{\bar{z}+\frac{1}{z+\frac{1}{\bar{z}}}},$$

where $\bar{z}$ is the complex conjugate of $z$. Determine the geometric form that satisfies

$$\operatorname{Re}(\zeta)=\operatorname{Re}\left(\frac{z+\frac{1}{4}}{|z|^{2}}\right)$$

Find solutions to

$$\operatorname{Im}(\log z)=\frac{\pi}{3}$$

and

$$z^{2}=x^{2}-y^{2}+2 i x,$$

where $z=x+i y$ is a complex variable. Sketch these solutions in the complex plane and describe the region they enclose. Derive complex equations for the circumscribed and inscribed circles for the region. [The circumscribed circle is the circle that passes through the vertices of the region and the inscribed circle is the largest circle that fits within the region.]