Netherlands — Second Round

Given is the sequence of numbers $a_0, a_1, a_2, \dots, a_{2020}$ with $a_0 = 0$. Furthermore, the following holds for every $k = 1, 2, \dots, 2020$:
$$a_k = \begin{cases} a_{k-1} \cdot k & \text{if } k \text{ is divisible by } 8, \\ a_{k-1} + k & \text{if } k \text{ is not divisible by } 8. \end{cases}$$
What are the last two digits of $a_{2020}$?