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E. Tree Constructing

time limit per test

4 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputYou 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

YES

3 1

4 1

1 2

5 2

2 6

Input

6 2 3

Output

NO

Input

10 4 3

Output

YES

2 9

2 10

10 3

3 1

6 10

8 2

4 3

5 6

6 7

Input

8 5 3

Output

YES

2 5

7 2

3 7

3 1

1 6

8 7

4 3

Codeforces (c) Copyright 2010-2018 Mike Mirzayanov

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