HMMT 2022 A5

Given a positive integer $k$, let $\|k\|$ denote the absolute difference between $k$ and the nearest perfect square. For example, $\|13\|=3$ since the nearest perfect square to 13 is 16 . Compute the smallest positive integer $n$ such that

$$\frac{\|1\|+\|2\|+\cdots+\|n\|}{n}=100 .$$