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This is Open [Un]orthodox/Conservative method:
1) Decide what conclusion you want to arrive at. This will often be based on predicting what the Jewish ethical response must be in a world that has changed so significantly from the early legal texts of Judaism, that the modern author is given much leeway.
4) Find a medrash as a springboard to show how quintessentially Jewish, how much in the spirit of Jewish law your own conclusion is.
5) Accept your original argument.
This would be The Center-left orthodox method:
A stylized version goes somewhat like this:
1) According to the Rambam, the Rosh, and the Kol Bo, activity X is proscribed. The Mordechai takes an even stricter view than the others, seeing multiple issurim involved.
2) No one explicitly permits it. However, a responsum by the Nodah Be-Yehudah includes an argument by his interlocutor which presents a lenient line of reasoning. The Nodah Be-Yehudah himself rejects it, with cause.
5) Therefore accept the lenient opinion.