Let \( \bar{z} \) denote the complex conjugate of a complex number \( z \). If \( z \) is a non-...

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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 \( |z| \) ?
(a) \( \left(\frac{43+3 \sqrt{205}}{2}\right)^{\frac{1}{4}} \)
(b) \( \left(\frac{7+\sqrt{33}}{4}\right)^{\frac{1}{4}} \)
(c) \( \left(\frac{9+\sqrt{65}}{4}\right)^{\frac{1}{4}} \)
(d) \( \left(\frac{7+\sqrt{13}}{6}\right)^{\frac{1}{4}} \)
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