For the non-homogeneous recurrence $a_n = 3a_{n-1} - 2a_{n-2} + 2^n$, the homogeneous part has characteristic equation $r^2 - 3r + 2 = 0$ with roots $r = 1$ and $r = 2$.

Since $2$ is a characteristic root, what form should the particular solution $a_n^{(p)}$ take?