Recall that a partition of a positive integer $n$ means a way of writing $n$ as the sum of some positive integers, where the order of the parts does not matter. For example, there are five partitions of $4$: $$4\qquad 3 1\qquad 2 2\qquad 2 1 1\qquad 1 1 1 1$$ How many partitions of $12$ are there that have at least four parts, such that the largest, second-largest, third-largest, and fourth-largest parts are respectively greater than or equal to $4,3,2,1$