An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence
\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \]for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence
\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \]for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
3
An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence
\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \]for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.
train