Linear Recurrence With Little Information

An infinite sequence of real numbers \(a_1,a_2,\ldots\) 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 \(a_1+a_2+\cdots+a_{100}\).


Source: HMMT.
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