Consider the coupled recurrence system: $\begin{cases} p_n = 2q_{n-1} \\ q_n = p_{n-1} + q_{n-1} \end{cases}$ with $p_0 = 1, q_0 = 1$. In matrix form, $\begin{pmatrix} p_n \\ q_n \end{pmatrix} = M^n \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ for some matrix $M$. What is the sum $p_3 + q_3$?