CF550D.Regular Bridge

An undirected graph is called k-regular, if the degrees of all its vertices are equal k. An edge of a connected graph is called a bridge, if after removing it the graph is being split into two connected components.

Build a connected undirected k-regular graph containing at least one bridge, or else state that such graph doesn't exist.

无向图被称为 k-正则图（k-regular），如果其所有顶点的度数均等于 k。连通图的一条边被称为桥（bridge），如果删除该边后，图将分裂为两个连通分量。

请构造一个包含至少一条桥的连通无向 k-正则图；若不存在这样的图，则说明其不存在。

The single line of the input contains integer k (1 ≤ k ≤ 100) — the required degree of the vertices of the regular graph.

输入仅包含一个整数 $k$（$1 \leq k \leq 100$）—— 所需正则图中各顶点的度数。

Print "NO" (without quotes), if such graph doesn't exist.

Otherwise, print "YES" in the first line and the description of any suitable graph in the next lines.

The description of the made graph must start with numbers n and m — the number of vertices and edges respectively.

Each of the next m lines must contain two integers, a and b (1 ≤ a, b ≤ n, a ≠ b), that mean that there is an edge connecting the vertices a and b. A graph shouldn't contain multiple edges and edges that lead from a vertex to itself. A graph must be connected, the degrees of all vertices of the graph must be equal k. At least one edge of the graph must be a bridge. You can print the edges of the graph in any order. You can print the ends of each edge in any order.

The constructed graph must contain at most 106 vertices and 106 edges (it is guaranteed that if at least one graph that meets the requirements exists, then there also exists the graph with at most 106 vertices and at most 106 edges).

如果这样的图不存在，则输出 "NO"（不带引号）。

否则，第一行输出 "YES"，随后几行输出任意一个满足条件的图的描述。

所构造图的描述必须以两个数 n 和 m 开头——分别表示顶点数和边数。

接下来的 m 行中，每行包含两个整数 a 和 b（1 ≤ a, b ≤ n，且 a ≠ b），表示存在一条连接顶点 a 和 b 的边。图中不得包含重边或自环。图必须是连通的，且所有顶点的度数均恰好为 k。图中至少有一条边是桥边（bridge）。你可以以任意顺序输出各条边；对每条边，其两个端点也可按任意顺序输出。

所构造的图至多包含 $10^6$ 个顶点和 $10^6$ 条边（题目保证：若至少存在一个满足要求的图，则必存在一个顶点数不超过 $10^6$、边数也不超过 $10^6$ 的满足要求的图）。

In the sample from the statement there is a suitable graph consisting of two vertices, connected by a single edge.

在题面的样例中，存在一个合适的图，该图由两个顶点通过一条边连接而成。