Points, Lines and Ready-made Titles

题意翻译

在一个平面上有n个不同的点（n给出）。而对于每一个点而言，你可以对它作一条垂直线，也可以作一条水平线，并且也可以不对它进行任何操作。（若有重合的线段则视为一条直线）。请问你能得到多少种不同的图片呢？得出的答案对1000000007（10^9+7）取模。

题目描述

You are given $ n $ distinct points on a plane with integral coordinates. For each point you can either draw a vertical line through it, draw a horizontal line through it, or do nothing. You consider several coinciding straight lines as a single one. How many distinct pictures you can get? Print the answer modulo $ 10^{9}+7 $ .

输入输出格式

输入格式

The first line contains single integer $ n $ ( $ 1<=n<=10^{5} $ ) — the number of points. $ n $ lines follow. The ( $ i+1 $ )-th of these lines contains two integers $ x_{i} $ , $ y_{i} $ ( $ -10^{9}<=x_{i},y_{i}<=10^{9} $ ) — coordinates of the $ i $ -th point. It is guaranteed that all points are distinct.

输出格式

Print the number of possible distinct pictures modulo $ 10^{9}+7 $ .

输入输出样例

输入样例 #1

4
1 1
1 2
2 1
2 2

输出样例 #1

16

输入样例 #2

2
-1 -1
0 1

输出样例 #2

9

说明

In the first example there are two vertical and two horizontal lines passing through the points. You can get pictures with any subset of these lines. For example, you can get the picture containing all four lines in two ways (each segment represents a line containing it). The first way:  The second way: In the second example you can work with two points independently. The number of pictures is $ 3^{2}=9 $ .