Score : $700$ points

Problem Statement

We have an undirected graph with $N$ vertices numbered $0$ through $N-1$ and no edges at first. You may add edges to this graph as you like. When you are done with adding edges, the following operation will be performed once using a given integer $K$.

• For each edge you have added, let $u$ and $v$ be its endpoints, and an edge will be added between two vertices $(u+K)$ $\mathrm{mod}$ $N$ and $(v+K)$ $\mathrm{mod}$ $N$. This edge will be added even if there is already an edge between these vertices, resulting in multi-edges.

For the given $N$ and $K$, find a set of edges that you should add so that the graph will be a tree after the operation. If there is no such set of edges, indicate that fact.

Constraints

• $2\leq N\leq 2\times 10^5$
• $1\leq K\leq N-1$
• All values in the input are integers.

Input

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

$N$ $K$

Output

If there is a set of edges that satisfies the requirement, let $M$ be the number of edges, and $a_i$ and $b_i$ be the two vertices connected by the $i$-th edge, and print a solution in the following format. Here, $0\leq M\leq N$ must hold, and all values must be integers. The edges may be printed in any order, as well as the endpoints of an edge.

$M$

$a_1$ $b_1$

$a_2$ $b_2$

$\vdots$

$a_M$ $b_M$

If multiple solutions exist, any of them will be accepted.

If no set of edges satisfies the requirement, print -1.

5 2

2
2 3
2 4

The operation will add the edges $4$-$0$ and $4$-$1$. Then, the graph will be a tree, so this is a legitimate output.

2 1

-1

There is no way to have a tree after the operation.

7 1

3
0 1
2 3
4 5