CMO 2021 Problem 1

Let $\{ z_n \}_{n \ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$ , $|z_k z_{k+1}|=2^k$ . Denote $f_n=|z_1+z_2+\cdots+z_n|,$ for $n=1,2,\cdots$ (1) Find the minimum of $f_{2020}$ . (2) Find the minimum of $f_{2020} \cdot f_{2021}$ .