A8871 | Ksusha and Square

Ksusha is a vigorous mathematician. She is keen on absolutely incredible mathematical riddles.

Today Ksusha came across a convex polygon of non-zero area. She is now wondering: if she chooses a pair of distinct points uniformly among all integer points (points with integer coordinates) inside or on the border of the polygon and then draws a square with two opposite vertices lying in the chosen points, what will the expectation of this square's area be?

A pair of distinct points is chosen uniformly among all pairs of distinct points, located inside or on the border of the polygon. Pairs of points $p,q$ $(p≠q)$ and $q,p$ are considered the same.

Help Ksusha! Count the required expectation.

The first line contains integer $n$ $(3<=n<=10^{5})$ — the number of vertices of Ksusha's convex polygon. Next $n$ lines contain the coordinates of the polygon vertices in clockwise or counterclockwise order. The $i$ -th line contains integers $x_{i},y_{i}$ $(|x_{i}|,|y_{i}|<=10^{6})$ — the coordinates of the vertex that goes $i$ -th in that order.

Print a single real number — the required expected area.

The answer will be considered correct if its absolute and relative error doesn't exceed $10^{-6}$ .