Find all integers $n \geq 8$ such that there exists a simple graph with $n$ vertices whose degrees are as follows:

(i) $(n-4)$ vertices of the graph are with degrees $4, 5, 6, \dots, n-2, n-1$, respectively.

(ii) The other $4$ vertices are with degrees $n-2, n-2, n-1, n-1$, respectively.