Let $x_0,x_1,x_2,\dots$ 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 $3S_n\equiv 1(\mathrm{mod}{2}^n)$ for all $n\ge 1$. What is the value of the sum $x_{2022}+2x_{2023}+4x_{2024}+8x_{2025}$?