The sequence $a_1,$ $a_2,$ $a_3,$ $\dots$ satisfies $a_1 = 19,$ $a_9 = 99,$ and for all $n \ge 3,$ $a_n$ is the arithmetic mean of the first $n - 1$ terms.  Find $a_2.$

Let's think step by step and output the final answer within \boxed{}.
user