Origami

Description:

Input:

The first line contains two integers $$$n$$$, $$$q$$$ ($$$3\le n\le 10^5, 1\le q\le 10^5$$$)  — the number of polygon vertices and queries, respectively.

Each of the next $$$n$$$ lines contains two integers $$$x_i$$$, $$$y_i$$$ ($$$|x_i|, |y_i|\le 10^5$$$)  — the coordinates of the $$$i$$$-th point of the polygon. The polygon has an edge connecting each pair of adjacent points in the input, and also an edge between $$$(x_1,y_1)$$$ and $$$(x_n,y_n)$$$. It is guaranteed that the polygon is non-degenerate and that any horizontal line intersects the boundary of the polygon in at most two points. In particular, no boundary edge is strictly horizontal. Two adjacent sides may be collinear.

Each of the next $$$q$$$ lines contains a single integer $$$f_i$$$ ($$$\min\limits_{j=1}^n(x_j)< f_i< \max\limits_{j=1}^n(x_j)$$$)  — the $$$x$$$-coordinate of the $$$i$$$-th fold query. It is guaranteed that all $$$f_i$$$ are distinct.

Output:

For each query, output the area $$$A_i$$$ of the paper if you fold it along the line $$$x=f_i$$$.

Your answer is considered correct if its absolute or relative error does not exceed $$$10^{-4}$$$.

Formally, let your answer be $$$a$$$, and the jury's answer be $$$b$$$. Your answer is accepted if and only if $$$\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-4}$$$.

Sample Input:

4 7
0 10
10 0
0 -10
-10 0
-9
-5
-1
0
1
5
9

Sample Output:

199.0000000000
175.0000000000
119.0000000000
100.0000000000
119.0000000000
175.0000000000
199.0000000000

Sample Input:

4 1
0 0
0 2
2 4
2 2
1

Sample Output:

3.0000000000

Sample Input:

9 4
0 -3
2 -2
-1 -1
0 4
2 5
1 6
-2 3
-1 1
-3 0
0
-2
-1
1

Sample Output:

11.1250000000
11.7500000000
10.3446969697
11.3333333333

Note:

The first test, with the fold $$$f=-5$$$:

The second test, with the fold $$$f=1$$$:

The third test, with the fold $$$f=-1$$$: