Find the polynomial $q(x)$ with integer coefficients such that when $p(x) = x^4 + 3x^3 + 3x^2 + 3x + 1$ is divided by $x^2 - x + 1$, the remainder is $3x + 1$. Then, identify the degree of $q(x)$.

### Let's think step by step and output the final answer within \boxed{}.
user