You are given a red-black tree T with 15 internal nodes (nodes that hold key values) that form a full binary tree of height 3 (i.e., a full binary tree of height 4 if you include the NIL leaves). Can you assign colors to the nodes so that a call to RB-INSERT(T, z) for any new key value z.key will cause RB-INSERT-FIxUP(T, 2) to change the color of the root to red before switching it back to black? The initial assignment of colors needs to obey the red-black properties. If such a color assignment exists, then provide a sequence of 15 numbers whose insertion (in that order) would lead to such a tree, along with a figure of the resulting tree. If not, then explain why such an assignment cannot exist, using the fact that the tree needs to satisfy the red-black properties