Tree Constructing

You are given three integers $ n $ , $ d $ and $ k $ . Your task is to construct an undirected tree on $ n $ vertices with diameter $ d $ and degree of each vertex at most $ k $ , or say that it is impossible. An undirected tree is a connected undirected graph with $ n - 1 $ edges. Diameter of a tree is the maximum length of a simple path (a path in which each vertex appears at most once) between all pairs of vertices of this tree. Degree of a vertex is the number of edges incident to this vertex (i.e. for a vertex $ u $ it is the number of edges $ (u, v) $ that belong to the tree, where $ v $ is any other vertex of a tree).

The first line of the input contains three integers $ n $ , $ d $ and $ k $ ( $ 1 \le n, d, k \le 4 \cdot 10^5 $ ).

If there is no tree satisfying the conditions above, print only one word "NO" (without quotes). Otherwise in the first line print "YES" (without quotes), and then print $ n - 1 $ lines describing edges of a tree satisfying the conditions above. Vertices of the tree must be numbered from $ 1 $ to $ n $ . You can print edges and vertices connected by an edge in any order. If there are multiple answers, print any of them.

6 3 3

YES
3 1
4 1
1 2
5 2
2 6

6 2 3

NO

10 4 3

YES
2 9
2 10
10 3
3 1
6 10
8 2
4 3
5 6
6 7

8 5 3

YES
2 5
7 2
3 7
3 1
1 6
8 7
4 3