HMMT 2023 A6

Suppose $a_{1}, a_{2}, \ldots, a_{100}$ are positive real numbers such that

$$a_{k}=\frac{k a_{k-1}}{a_{k-1}-(k-1)}$$

for $k=2,3, \ldots, 100$. Given that $a_{20}=a_{23}$, compute $a_{100}$.