Let $x,y,z$ be real numbers. Solve the following system: $$\left\{\begin{matrix} x^{100}+\frac {1}{x^{100}}=y^{2018}+1 \\ y^{1918}+\frac {1}{y^{1918}}=z^{2018}+1 \\ z^{100}+\frac {1}{z^{100}}=x^{2018}+1 \end{matrix}\right.$$
What I have tried is to consider two cases: $x\geq y \geq z\geq 1$ and then prove that $x=y=z=1$. And the other case $x \leq y \leq z \leq -1$ and then prove $x=y=z=-1$. But I can't find a link between the variables. Any ideea or help I would greatfully appreciate.