I need to solve for 4 unknowns $x_{1}, x_{2}, x_{3}, x_{4}$ from 4 polynomial equations, three quadratic and one linear, given by the system
$$ (1-2c_{1})x_{1} - (1-2c_{2})x_{2} + c_{4}x_{2}x_{3} + (c_{2} - c_{1})x_{3}x_{4} + c_{3}x_{2}x_{4} - c_{4}x_{3}x_{1} - c_{3}x_{1}x_{4} = 0, \qquad (1)\\ (1-2c_{2})x_{2} - (1-2c_{3})x_{3} - c_{4}x_{1}x_{2} - c_{1}x_{2}x_{4} + c_{4}x_{1}x_{3} + (c_{3}-c_{2})x_{1}x_{4} + c_{1}x_{3}x_{4} = 0, \qquad (2)\\ (1-2c_{3})x_{3} - (1-2c_{4})x_{4} + (c_{4}-c_{3})x_{1}x_{2} - c_{1}x_{2}x_{3} - c_{2}x_{1}x_{3} + c_{1}x_{2}x_{4} + c_{2}x_{1}x_{4} = 0, \qquad (3)\\ \hspace{4.49in} x_{1} + x_{2} + x_{3} + x_{4} = 1, \qquad (4)$$
where the coefficients $c_{i} \in (0, \frac{1}{2})$, $\sum_{i=1}^{4}c_{i}=1$, and the unknowns $x_{i} \in (0,1)$ for $i=1,...,4$. Is there a systematic way to compute the $x_{i}$'s as explicit functions of $c_{i}$'s?
This system is a special case of a more general system, where some analysis shows that there is a unique solution under the stated assumptions on $c_i$'s and $x_i$'s. I am interested to explicitly find the solution in this 4 unknown case.