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 $x^2+x+2$, the remainder is $3x-1$. 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?