"Nobody had a clear idea for what or where they supposed to search."
Does this construction look natural for natives?
I am trying to avoid ending the sentence with a preposition that would incorrectly modify also the adverb where.
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Nobody had a clear idea for what or where they supposed to search.
This is fine, except that there is a small chance it could be interpreted as:
Nobody had a clear idea for what or for where they supposed to search
Nobody had a clear idea where or for what they supposed to search
is a bit clearer.
Nobody had a clear idea where or what they supposed to search for.
there's little to no chance that anyone would attach the for to where. But it still sounds a bit awkward because it sounds like you tacked on the words "or what they were supposed to search for" without much forethought - which is fine in speech, but in writing, you have the time to think about it.