\[\large a! + b! + c! = a! \cdot b!\]

If \(a,b,c\) are nonnegative integers, then there are a finite number of solutions to the above equation. Let them be \((a_1, b_1, c_1), (a_2, b_2, c_2), (a_3, b_3, c_3), \ldots, (a_n, b_n, c_n)\) in some order. Find the value of

\[n + \displaystyle \sum_{i=1}^n (a_i+b_i+c_i ).\]

(If \(n = 0\), then the sum is zero, so you should write 0 as the answer.)

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