Problem Statement

You are given an integer $N$. Determine if there exists a tree with $2N$ vertices numbered $1$ to $2N$ satisfying the following condition, and show one such tree if the answer is yes.

• Assume that, for each integer $i$ between $1$ and $N$ (inclusive), Vertex $i$ and $N+i$ have the weight $i$. Then, for each integer $i$ between $1$ and $N$, the bitwise XOR of the weights of the vertices on the path between Vertex $i$ and $N+i$ (including themselves) is $i$.

Constraints

• $N$ is an integer.
• $1 \leq N \leq 10^{5}$

Input

Input is given from Standard Input in the following format:

N

Output

If there exists a tree satisfying the condition in the statement, print Yes; otherwise, print No. Then, if such a tree exists, print the $2N-1$ edges of such a tree in the subsequent $2N-1$ lines, in the following format:

a_{1} b_{1}
\vdots
a_{2N-1} b_{2N-1}

Here each pair ($a_i$, $b_i$) means that there is an edge connecting Vertex $a_i$ and $b_i$. The edges may be printed in any order.