4000 Followers Problem!

Number Theory Level 5

$\large a! + b! + c! = a! \cdot b!$

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.)