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First, I wouldn't tell them as once you file your application they will know you are going thru. So you won't "surprise them during recruitment."
As for your second question, it could. I know that in some places with some groups, they do not like having an RA in their own group. It can cause conflict of interest issues for some - it does for my group - as your first loyalty should be to your group's rules but in your case it's to the University's. Might be a problem. But that's up to each individual group.
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