"Surprise!" said c

a!b!=a!+b!+c2\displaystyle \large a!b! = a! + b! + c^2

Positive integers aa, bb, and cc, with a>ba > b, are the unique solution to the equation above.

If abc=mn\dfrac {a^b}{c} = \dfrac mn for coprime positive integers mm and nn, find m+nm + n.

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