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I agree with what you've done so far.  Now you just need to write S_b in base b.    It will help to break things up into cases where b is even or b is odd.  

And these identities might also help.

b^(4)/2 + b^(2)/2 = b^(3)(b/2) + b(b/2)

b^(4)/2 + b^(2)/2 = b^(3)(b-1)/2 + b^(2)(b+1)/2

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