Factorial!

Let $m$ and $n$ be positive integers satisfying $n! + 76 = m^2.$

If all the solutions of $(m,n)$ are $(m_1, n_1) , (m_2, n_2) , \ldots , (m_k , n_k )$ , submit your answer as $\displaystyle \sum_{j=1}^k (m_j + n_j)$ .

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Notation: $!$ is the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .