The starting-point, widely recognized, is that there were Western Asiatic people in Egypt. Call them: Asiatics, Semites, Canaanites, Levantine peoples. These aliens were there, for centuries, and they were coming and going all along, just not in millions at a time.Background here.
The second step was to identify a group among these as the Levites. They are the ones in Israel with Egyptian names, the ones who foster circumcision, a known practice in Egypt, the ones with connections to Egyptian material culture: a Tabernacle that has parallels with the battle tent of Rameses II, an ark that has parallels with Egyptian barks.
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