Well done! Sounds like it's going well with the "more than alibi sentences" thing so far. :-)
I'm also more likely to get interesting stories out of fandom-specific prompts, which is why most of my published fanfics have been written for exchanges etc. But I also like generic prompts, because thinking about them in connection with my fandoms can give me stories nobody happens to be requesting with those fandoms/characters. And for my original writing, prompts of all kinds can be really helpful.
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I'm also more likely to get interesting stories out of fandom-specific prompts, which is why most of my published fanfics have been written for exchanges etc. But I also like generic prompts, because thinking about them in connection with my fandoms can give me stories nobody happens to be requesting with those fandoms/characters. And for my original writing, prompts of all kinds can be really helpful.