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

Let's think step by step and output the final answer within \boxed{}.
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