Second Round, March 2019

If you try to divide the number $19$ by $5$, you will get a remainder. The number $5$ fits $3$ times in $19$ and you will be left with $4$ as remainder. There are two positive integers $n$ having the following property: if you divide $n^2$ by $2n+1$, you will get a remainder of $1000$. What are these two integers?