Not a teacher, but a fellow learner.
My humble opinion is that it's possible. We don't have to stick to the clause-only idea after "which". Even a phrase can come after "which". A similar example would be "It was a very powerful weapon with which to kill people."
"It was an awkward question with which to be awakened."
I think this is a habit of some eastern Asian languages - Koreans do the same thing. If you said "How do you think...(Ni zenme juede?)
", it wouldn't mean the content of your thinking, but the way you think such as thinking(deeply, standing, fast, etc)
HowWhat do you think of it?
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