KZ-MO 2021 Problem 3

Let $(a_n)$ and $(b_n)$ be sequences of real numbers, such that $a_1 = b_1 = 1$ , $a_{n+1} = a_n + \sqrt{a_n}$ , $b_{n+1} = b_n + \sqrt[3]{b_n}$ for all positive integers $n$ . Prove that there is a positive integer $n$ for which the inequality $a_n \leq b_k < a_{n+1}$ holds for exactly 2021 values of $k$ .