Teaser 2953: Marble tower

From The Sunday Times, 28th April 2019 [link]

Liam has a number of bags of marbles; each bag contains the same number (more than 1) of equal-size marbles.

He is building a tetrahedron with the marbles, starting with a layer which fits snugly in a snooker triangle. Each subsequent triangular layer has one fewer marble along each edge. With just one bag left he had completed a whole number of layers; the number of marbles along the edge of the triangle in the last completed layer was equal to the number of completed layers. The last bag had enough marbles to just complete the next layer.

How many bags of marbles did Liam have?

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