HMMT November 2020 U25

Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of positive integers where $a_{1}=\sum_{i=0}^{100} i$ ! and $a_{i}+a_{i+1}$ is an odd perfect square for all $i \geq 1$. Compute the smallest possible value of $a_{1000}$.