E. Tree Constructing
time limit per test
4 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

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).

Input

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

Output

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.1

Examples
Input
6 3 3
Output
YES3 14 11 25 22 6
Input
6 2 3
Output
NO
Input
10 4 3
Output
YES2 92 1010 33 16 108 24 35 66 7
Input
8 5 3
Output
YES2 57 23 73 11 68 74 3