"Surprise!" said c

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

Positive integers \(a\), \(b\), and \(c\), with \(a > b\), are the unique solution to the equation above.

If \(\dfrac {a^b}{c} = \dfrac mn\) for coprime positive integers \(m\) and \(n\), find \(m + n\).

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