$a+\frac{1}{b}=b+\frac{1}{c}\text{ and }b+\frac{1}{c}=c+\frac{1}{a}$

$\left\{\begin{matrix}b=-\frac{-\sqrt{1+\left(ac\right)^{2}+4c^{2}-2ac}-ac+1}{2c}\text{, }&c\neq 0\text{ and }-\frac{-\sqrt{a^{2}c^{2}-2ac+4c^{2}+1}-ac+1}{2c}=c+\frac{1}{a}-\frac{1}{c}\text{ and }a\neq 0\\b=-\frac{\sqrt{1+\left(ac\right)^{2}+4c^{2}-2ac}-ac+1}{2c}\text{, }&c\neq 0\text{ and }-\frac{\sqrt{a^{2}c^{2}-2ac+4c^{2}+1}-ac+1}{2c}=c+\frac{1}{a}-\frac{1}{c}\text{ and }a\neq 0\end{matrix}\right.$