2022 AMC 10B Problem 25

Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$ . For each positive integer $n$ , define $$S_n = \sum_{k=0}^{n-1} x_k 2^k$$ Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$ . What is the value of the sum $$x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?$$