DAPC24-084.Human Pyramid

As chairman of the Building A Pyramid Committee, you are specialized in breaking one specific world record: building the highest human pyramid. Unfortunately, you only know a limited number of people who are willing to be in the pyramid. After all, building human pyramids does not make much money, so most people are volunteers.

A full human pyramid of height $h$ consists of $h$ layers of people. As seen from below, it has $h$ people on the first layer, $h-1$ on the second, $h-2$ on the third, and so forth until eventually the final layer has just a single person. To determine whether you can break the world record, you need to know how high a pyramid you can build. Given how many people are available, how tall is the highest possible human pyramid that these people can make?