Here "sin" represents the false god the people worship in Samaria. AT: "by the false god of Samaria" (See: [[rc://en/ta/man/translate/figs-metonymy]])
# As your god lives, Dan
This is a way of making a solemn oath. The people declare that they believe the god of Dan is certainly alive to emphasize that they will certainly do what they promise to do.
# As the way to Beersheba exists
This is probably a reference to the roads that pilgrims would take to Beersheba in order to worship idols there. Again, this is a way of making a solemn oath. They state that the way to Beersheba certainly exists in order to emphasize that they will certainly do what they promise to do.
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