Let $x_0,x_1,x_2,\dots$ be a infinite sequence of real numbers, such that the following three equalities are true: I- $x_{2k}=(4x_{2k-1}-x_{2k-2})^2$, for $k\geq 1$ II- $x_{2k+1}=|\frac{x_{2k}}{4}-k^2|$, for $k\geq 0$ III- $x_0=1$ a) Determine the value of $x_{2022}$ b) Prove that there are infinite many positive integers $k$, such that $2021|x_{2k+1}$