Under the secrecy ban, even Aldrich College cannot openly open a branch of mystery. Similarly, professors who enroll mystery related courses must also cover up.
Therefore, the so-called Mystery Academy does not exist in the open, at least, and the professors of Mystery Academy also have their own superficial identities.
For example, Merlin, whose apparent identity is the newly recruited professor of history department, is not entirely false. Merlin also needs to give some history lessons to ordinary students, so that his position as a professor of history department will not be doubted.
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