I wonder whether the system of equations and inequations below have a solution. If there are solutions, what are they? A numerical solution is also desired.

$\{\begin{array}{l}\frac{{c}_{1}}{1-{x}_{1}}+\frac{{c}_{2}}{1-{x}_{2}}+\frac{{c}_{3}}{1-{x}_{3}}=0\\ \frac{{c}_{1}}{1-{x}_{4}}+\frac{{c}_{2}}{1-{x}_{5}}+\frac{{c}_{3}}{1-{x}_{6}}=0\\ {c}_{1}\mathrm{ln}\frac{{x}_{1}}{1-{x}_{1}}+{c}_{2}\mathrm{ln}\frac{{x}_{2}}{1-{x}_{2}}+{c}_{3}\mathrm{ln}\frac{{x}_{3}}{1-{x}_{3}}=0\\ {c}_{1}\mathrm{ln}\frac{{x}_{4}}{1-{x}_{4}}+{c}_{2}\mathrm{ln}\frac{{x}_{5}}{1-{x}_{5}}+{c}_{3}\mathrm{ln}\frac{{x}_{6}}{1-{x}_{6}}=0\\ \frac{{x}_{1}(1-{x}_{1})}{{x}_{4}(1-{x}_{4})}=\frac{{x}_{2}(1-{x}_{2})}{{x}_{5}(1-{x}_{5})}=\frac{{x}_{3}(1-{x}_{3})}{{x}_{6}(1-{x}_{6})}>1\end{array}$

where ${c}_{1},{c}_{2},{c}_{3}$ are constants, and ${x}_{i}\in (0,1),i=1,2,3,4,5,6$

$\{\begin{array}{l}\frac{{c}_{1}}{1-{x}_{1}}+\frac{{c}_{2}}{1-{x}_{2}}+\frac{{c}_{3}}{1-{x}_{3}}=0\\ \frac{{c}_{1}}{1-{x}_{4}}+\frac{{c}_{2}}{1-{x}_{5}}+\frac{{c}_{3}}{1-{x}_{6}}=0\\ {c}_{1}\mathrm{ln}\frac{{x}_{1}}{1-{x}_{1}}+{c}_{2}\mathrm{ln}\frac{{x}_{2}}{1-{x}_{2}}+{c}_{3}\mathrm{ln}\frac{{x}_{3}}{1-{x}_{3}}=0\\ {c}_{1}\mathrm{ln}\frac{{x}_{4}}{1-{x}_{4}}+{c}_{2}\mathrm{ln}\frac{{x}_{5}}{1-{x}_{5}}+{c}_{3}\mathrm{ln}\frac{{x}_{6}}{1-{x}_{6}}=0\\ \frac{{x}_{1}(1-{x}_{1})}{{x}_{4}(1-{x}_{4})}=\frac{{x}_{2}(1-{x}_{2})}{{x}_{5}(1-{x}_{5})}=\frac{{x}_{3}(1-{x}_{3})}{{x}_{6}(1-{x}_{6})}>1\end{array}$

where ${c}_{1},{c}_{2},{c}_{3}$ are constants, and ${x}_{i}\in (0,1),i=1,2,3,4,5,6$