Find all positive unordered multi-sets of integers \((a,b,c)\) such that \({ab-c,bc-a,ca-b}\) are powers of 2.

If these sets can be represented as \[(a_1,b_1,c_1),(a_2,b_2,c_2),\dots,(a_n,b_n,c_n).\] Evaluate \(\displaystyle\large{\left(\sum_{i=1}^{n} a_i + b_i + c_i\right) +n}\).

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