Suppose $a_1, a_2, ... , a_{100}$ are positive real numbers such that $$a_k =\frac{ka_{k-1}}{a_{k-1} - (k - 1)}$$ for $k = 2, 3, ... , 100$. Given that $a_{20} = a_{23}$, compute $a_{100}$.


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