How does Gemini have Lucy's keys?
Gemini has the power to copy things. So since Gemini can copy the keys doesn't mean that lucy would be able to use them. Duplicating doesn't mean to copy. Copying something like on a printer isn't the real design. It's a "copy" of the real thing. Meaning that she wouldn't be able to use the copy, but she would be able to use the duplicate. Saying that Gemini only uses copies not duplicates. So lucy would not be able to use the fake keys.