HMMT 2016 A5

An infinite sequence of real numbers $a_{1}, a_{2}, \ldots$ satisfies the recurrence

$$a_{n+3}=a_{n+2}-2 a_{n+1}+a_{n}$$

for every positive integer $n$. Given that $a_{1}=a_{3}=1$ and $a_{98}=a_{99}$, compute $a_{1}+a_{2}+\cdots+a_{100}$. Proposed by: Evan Chen