BALTIC-WAY 2023 Problem 3

Denote a set of equations in the real numbers with variables $x_{1}, x_{2}, x_{3} \in \mathbb{R}$ Flensburgian if there exists an $i \in\{1,2,3\}$ such that every solution of the set of equations where all the variables are pairwise different, satisfies $x_{i}>x_{j}$ for all $j \neq i$.

Determine for which positive integers $n \geq 2$, the following set of two equations

$$a^{n}+b=a \text { and } c^{n+1}+b^{2}=a b$$

in the three real variables $a, b, c$ is Flensburgian.