A recurrence with characteristic equation $(r-2)^3 = 0$ has a triple root at $r=2$. The general solution is $a_n = (c_1 + c_2 n + c_3 n^2) \cdot 2^n$. Given $a_0 = 1, a_1 = 4, a_2 = 16$, find $a_3$.