The sequence $(a_n)$ is defined by $a_1 = 1,$ $a_2 = 2,$ and
\[a_n^2 - a_{n - 1} a_{n + 1} = 1\]for all $n \ge 2.$  Find $a_{100}.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
The sequence $(a_n)$ is defined by $a_1 = 1,$ $a_2 = 2,$ and
\[a_n^2 - a_{n - 1} a_{n + 1} = 1\]for all $n \ge 2.$  Find $a_{100}.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
100
The sequence $(a_n)$ is defined by $a_1 = 1,$ $a_2 = 2,$ and
\[a_n^2 - a_{n - 1} a_{n + 1} = 1\]for all $n \ge 2.$  Find $a_{100}.$
train