Number of ordered pairs

\[m^{2}+2(3^{n})=m(2^{n+1}-1)\] \(m,n\) are positive integers . \((m_{1},n_{1})......,(m_{ω},n_{ω})\) be the solutions to the above equation. Find \(ω+\sum_{i=1}^{ω}m_{i}+n_{i}\).

Problem is original.

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