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.