AMC 10 2022b Problem 21

Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x+2$, and when $P(x)$ is divided by the polynomial $x^2+1$, the remainder is $2x+1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial? $\textbf{(A)}\ 10 \textbf{(B)}\ 13 \textbf{(C)}\ 19 \textbf{(D)}\ 20 \textbf{(E)}\ 23$