Suppose there exists an algorithm that solves 3SAT in a polynomial number of steps. However, this algorithm requires some 'tuning' parameters, and the value of these tuning parameters take an exponential number of steps(given the input) to determine.
Would this place 3SAT in P? (since it can be solved in polynomial time given the appropriate parameters)
Or would 3SAT remain in NP? (since it still takes an exponential number of steps to ultimately solve)
Edit: just to clarify, the question is about whether or not 3SAT would 'move' to P or 'remain' where it currently is...NP. If 3SAT 'moves' to P it obviously would still be contained in NP.