So Many Possibilities!

Let \(S_{1}=[1,2,3]\). For any \(S_{n}=[a,b,c]\), define \(S_{n+1}=[b+c,b,c]\), that is, exactly one element from \(S_{n}\) is replaced with the sum of the other two elements. For how many values of \(n\) does \(S_{n}=[1234,2345,3456]\)?

For example, if \(S_{133}=[1234,2345,3456]\) and \(S_{583}=[1234,2345,3456]\), there would be two values of \(n\) that would contain the set \([1234,2345,3456]\), and your answer would be 2.

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