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The induction coil is retrieved from the top of the central machine in the ATLAS substructure during the quest Mother of Invention. The variant of the induction coil that is acquired depends on the option chosen when interacting with the machine. After acquiring the induction coil, the mysterious component will also pop out.
All of the quest items retrieved during the latter half of Mother of Invention, during the "Extract components from the ATLAS machines" objective and including the induction coil, are removed from the player character's inventory upon speaking with Scribe Valdez at the conclusion of the quest.
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