The first 2 quests were intro ones to Katheryn in Fontaine and new weapon designs, which need to be traded for. Then comes the important quest with Virgil and his not Liliacruces Forum but Institute of Natural Philosophy. The quest made it quite obvious at the beginning that he was not supposed to be trusted so the betrayal at the end was expected and he'll definitely show up later. Who is he though as he knew about the Institute, is against the current Fontaine as he is being hunted down, and is clearly plotting something with whatever the Institute was researching. They looked into machines, Oceanids and a bunch of other stuff which I probably should have paid more attention to.
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