L. Birthday bash
time limit per test
5 seconds
memory limit per test
1024 megabytes
input
standard input
output
standard output

Today is Alice's birthday! She invited you and her $$$N-2$$$ other friends to her birthday bash.

The birthday cake is a convex polygons with $$$N$$$ sides such that no two adjacent sides are parallel. Once they sing Happy Birthday, the cake is cut as follows: a point $$$P$$$ is uniformly chosen at random in the interior of the pizza; from point $$$P$$$, $$$N$$$ cuts are made to the corners cake creating $$$N$$$ triangular slices. Since you and Alice are nice, you let your $$$N-2$$$ friends first take a slice, then you take a slice, and Alice gets the last one.

If everyone greedily takes cake slices by area, what is the expected area of your slice?

Input

The first line contains a single integer $$$N$$$ ($$$3 \leq N \leq 200$$$).

The $$$i$$$-th of the following $$$N$$$ lines contains two integers $$$x_i$$$ and $$$y_i$$$: the coordinates of the $$$i$$$-th polygon vertex ($$$-10^5 \leq x_i, y_i \leq 10^5$$$). The $$$x$$$-axis runs from left to right, and the $$$y$$$-axis runs from bottom to top. The vertices are numbered in counterclockwise order.

Output

Print one real number: the expected area of your slice of cake. Your answer will be considered correct if its absolute or relative error does not exceed $$$10^{-6}$$$.

Examples
Input
4
0 0
1 0
1 1
0 1
Output
0.1666666667
Input
10
-43 33
-35 -39
32 -41
46 -12
50 30
50 34
49 38
42 49
37 49
-6 44
Output
95.7188121428