Problem Statement

We define the density of a non-empty simple undirected graph as $\displaystyle\frac{(\text{number\ of\ edges})}{(\text{number\ of\ vertices})}$.

You are given positive integers $N$ and $D$. Determine whether there is a simple undirected graph $G$ with $N$ vertices and $DN$ edges that satisfies the following condition. If it exists, find one such graph.

Condition: Let $V$ be the vertex set of $G$. For any non-empty proper subset $X$ of $V$, the density of the induced subgraph of $G$ by $X$ is strictly less than $D$.

What is an induced subgraph? For a vertex subset $X$ of graph $G$, the induced subgraph of $G$ by $X$ is the graph with vertex set $X$ and edge set containing all edges of $G$ that connect two vertices in $X$. In the above condition, note that we only consider vertex subsets that are neither empty nor the entire set.

Constraints

• $N \geq 1$
• $D \geq 1$
• $DN \leq 5 \times 10^4$

Input

The input is given from Standard Input in the following format:

N D

Output

If there is a simple undirected graph that satisfies the condition, print Yes; otherwise, print No. Furthermore, in the case of Yes, print a graph that satisfies the condition in the following format in the next $DN$ lines. If multiple graphs satisfy the condition, any of them will be considered correct.

A_1 B_1
A_2 B_2
\vdots
A_{DN} B_{DN}

• $1 \leq A_i, B_i \leq N \ \ (1 \leq i \leq DN)$
• $A_i \neq B_i \ \ (1 \leq i \leq DN)$
• $\{A_i, B_i\} \neq \{A_j, B_j\} \ \ (1 \leq i < j \leq DN)$
• The vertices of the graph are numbered from $1$ to $N$.
• The $i$-th line of the output represents that the $i$-th edge connects vertex $A_i$ and vertex $B_i$.
• The order of the edges (which edge is printed first) and the order of the vertices (which endpoint of each edge is printed first) may be arbitrary.