Binary or Septenary

Computer Science Level pending

Split the integers from 0 to 127 into two groups A and B such that sum of first, second, third, fourth, fifth and sixth powers of numbers in group A is equal to those in B, respectively.

\[\text{i.e.} \sum_{\forall A_i}A_i^{p}= \sum_{\forall B_i}B_i^{p} \quad\quad\forall p\in\{1,2,3,4,5,6\}\]

Find the difference between their seventh power sum. That is, find

\[\large\left|\sum_{\forall A_i}A_i^{7}- \sum_{\forall B_i}B_i^{7}\right|.\]

PS I don't know whether the solution I have is unique or not. So, if you get another solution and it satisfies conditions post it in this thread

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