CIIM 2020 Problem 5

Determine all the positive real numbers $x_1, x_2, x_3, \dots, x_{2021}$ such that $x_{i+1}=\frac{x_i^3+2}{3x_i^2}$ for every $i=1, 2, 3, \dots, 2020$ and $x_{2021}=x_1$