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 $\displaystyle\large{\left(\sum_{i=1}^{n} a_i + b_i + c_i\right) +n}$.