Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $(\bar{z})^{2}+\frac{1}{z^{2}}$ are integers, then which of the following is/are possible value(s) of $|z|$ ?

$\text{Given: } \bar{z} = \text{complex conjugate of } z,\quad z \neq 0$

$\operatorname{Re}\!\left((\bar{z})^{2}+\frac{1}{z^{2}}\right) \in \mathbb{Z}$

$\operatorname{Im}\!\left((\bar{z})^{2}+\frac{1}{z^{2}}\right) \in \mathbb{Z}$

$\text{To find: possible value(s) of } |z|$

$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$

$r=|z|>0,\quad \theta=\arg z$

$\bar{z} = r(\cos\theta - i\sin\theta) = re^{-i\theta}$

$|\bar{z}| = |z| = r$

$(\bar{z})^{2} = r^{2}e^{-2i\theta}$

$\frac{1}{z^{2}} = \frac{1}{r^{2}}\,e^{-2i\theta}$

$\text{Let } w = (\bar{z})^{2}+\frac{1}{z^{2}} = \left(r^{2}+\frac{1}{r^{2}}\right)e^{-2i\theta}$

$e^{-2i\theta} = \cos 2\theta - i\sin 2\theta$

$\operatorname{Re}(w) = \left(r^{2}+\frac{1}{r^{2}}\right)\cos 2\theta$

$\operatorname{Im}(w) = -\left(r^{2}+\frac{1}{r^{2}}\right)\sin 2\theta$

$m = \operatorname{Re}(w),\quad n = \operatorname{Im}(w),\quad m,n\in\mathbb{Z}$

$m^{2}+n^{2} = \left(r^{2}+\frac{1}{r^{2}}\right)^{2}(\cos^{2}2\theta+\sin^{2}2\theta)$

$\Rightarrow m^{2}+n^{2} = \left(r^{2}+\frac{1}{r^{2}}\right)^{2}$

$a = r^{2}+\frac{1}{r^{2}} \ge 2,\qquad a^{2} = m^{2}+n^{2}$

$a^{2}\ \text{is a sum of two squares of integers}$

$R = r^{4} = |z|^{4}\ \ (\text{the radicand in each option})$

$a^{2} = R+\frac{1}{R}+2$

$(1)\ R=\frac{43+3\sqrt{205}}{2},\quad \frac{1}{R}=\frac{43-3\sqrt{205}}{2}$

$a^{2}=R+\frac{1}{R}+2 = 43+2 = 45$

$45 = 6^{2}+3^{2}\ \Rightarrow\ \text{possible}$

$(2)\ a^{2}=\frac{7+\sqrt{33}}{4}+\frac{7-\sqrt{33}}{4}+2=\frac{7}{2}+2$

$a^{2}=\frac{11}{2}\notin\mathbb{Z}\ \Rightarrow\ \text{not possible}$

$(3)\ a^{2}=\frac{9+\sqrt{65}}{4}+\frac{9-\sqrt{65}}{4}+2=\frac{9}{2}+2$

$a^{2}=\frac{13}{2}\notin\mathbb{Z}\ \Rightarrow\ \text{not possible}$

$(4)\ a^{2}=\frac{7+\sqrt{13}}{6}+\frac{7-\sqrt{13}}{6}+2=\frac{7}{3}+2$

$a^{2}=\frac{13}{3}\notin\mathbb{Z}\ \Rightarrow\ \text{not possible}$

$(1):\ a=\sqrt{45},\ \cos 2\theta=\frac{6}{\sqrt{45}},\ \sin 2\theta=-\frac{3}{\sqrt{45}}$

$\boxed{|z|=\left(\frac{43+3\sqrt{205}}{2}\right)^{1/4}}$

$\text{Hence only option (1) is possible}$