For the recurrence $a_n = 5a_{n-1} - 6a_{n-2}$ with characteristic roots $r_1 = 2, r_2 = 3$, the general solution is $a_n = c_1 \cdot 2^n + c_2 \cdot 3^n$.

Given $a_0 = 1$ and $a_1 = 5$, find $a_3$.