Let $z$ be a complex number with non-zero imaginary part. If $\frac{2+3z+4z^{2}}{2-3z+4z^{2}}$ is a real number, then the value of $|z|^{2}$ is ______.

$\text{Given: } z \in \mathbb{C}, \ \operatorname{Im}(z) \neq 0$ $\dfrac{2+3z+4z^{2}}{2-3z+4z^{2}} \in \mathbb{R}$ $\text{To find: } |z|^{2}$ $w = \dfrac{2+3z+4z^{2}}{2-3z+4z^{2}}$ $A = 2+4z^{2}, \quad B = 3z$ $w = \dfrac{A+B}{A-B}$ $w = \bar w$ $\dfrac{A+B}{A-B} = \dfrac{\bar A+\bar B}{\bar A-\bar B}$ $(A+B)(\bar A-\bar B) = (\bar A+\bar B)(A-B)$ $A\bar A - A\bar B + B\bar A - B\bar B$ $= \bar A A - \bar A B + \bar B A - \bar B B$ $B\bar A = A\bar B$ $3z\,(2+4\bar z^{2}) = (2+4z^{2})\,3\bar z$ $2z + 4z\bar z^{2} = 2\bar z + 4z^{2}\bar z$ $2(z-\bar z) = 4z\bar z\,(z-\bar z)$ $\operatorname{Im}(z) \neq 0 \Rightarrow z \neq \bar z$ $1 = 2\,z\bar z = 2|z|^{2}$ $|z|^{2} = \dfrac{1}{2}$ $z = \tfrac{1}{2}+\tfrac{1}{2}i \Rightarrow |z|^{2} = \tfrac{1}{2} \; \checkmark$ $\boxed{|z|^{2} = \tfrac{1}{2}}$