Factorial and a Power of Two

\[\large a! b!=a!+b!+2^c\]

Let all the triplets of positive integer solutions \((a,b,c) \) satisfying the equation above be \((a_1, b_1, c_1) , (a_2, b_2, c_2) , \ldots , (a_n , b_n, c_n) \). Find \[ (a_1 + b_1 + c_1) + (a_2 + b_2+c_2) + \cdots + (a_n + b_n + c_n) . \]

\[\]Notation:
\(!\) denotes the factorial. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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