FKMO 2020 Problem 4

Do there exist two positive reals $\alpha, \beta$ such that each positive integer appears exactly once in the following sequence? $$2020, [\alpha], [\beta], 4040, [2\alpha], [2\beta], 6060, [3\alpha], [3\beta], \cdots$$ If so, determine all such pairs; if not, prove that it is impossible.