2022 AMC 10B Problem 25

Let x_0,x_1,x_2,\ldots 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}? \textbf{(A) } 6 \qquad \textbf{(B) } 7 \qquad \textbf{(C) }12\qquad \textbf{(D) } 14\qquad \textbf{(E) }15

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.