2021 AMC 12A Problem 16

In the following list of numbers, the integer $n$ appears $n$ times in the list for $1\le n \le 200$ . $1,2,2,3,3,3,4,4,4,\ldots,200,200,\ldots,200$ What is the median of the numbers in this list?

$\textbf{(A) }100.5 \qquad \textbf{(B) }134 \qquad \textbf{(C) }142 \qquad \textbf{(D) }150.5\qquad \textbf{(E) }167$

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.