Australian AMC 2022 - Senior Question 19

A sequence of values $a_1,a_2,\ldots,a_{100}$ is calculated as follows: $a_1=1$, $a_2=2$, $a_3=\frac{a_2+1}{a_1}$, $\ldots$, $a_n=\frac{a_{n-1}+1}{a_{n-2}}$, $\ldots$, $a_{100}=\frac{a_{99}+1}{a_{98}}$. What is $a_{100}$?