Middle European Mathematical Olympiad 2020 Problem I-4

Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \ldots, x_n$ such that

$$\dfrac{1}{x_1^2} + \dfrac{2}{x_2^2} + \dfrac{4}{x_3^2} + \cdots + \dfrac{2^{n-1}}{x_n^2} = 1.$$