CF293D.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.

库舒沙是一位充满活力的数学家，她热衷于解决那些令人难以置信的数学谜题。

今天，库舒沙遇到了一个面积非零的凸多边形。她现在思考这样一个问题：如果她在该多边形内部或边界上所有整点（即坐标均为整数的点）中，均匀随机地选取一对互异的点，然后以这对点为某正方形的一组对顶点作正方形，那么该正方形面积的期望值是多少？

所有位于多边形内部或边界上的互异点对被等概率选取；点对 $p,\,q$（其中 $p \ne q$）与 $q,\,p$ 被视为同一对。

请帮助库舒沙！计算该期望值。

The first line contains integer n (3 ≤ n ≤ 105) — 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| ≤ 106) — the coordinates of the vertex that goes i-th in that order.

第一行包含一个整数 $ n $（$ 3 \leq n \leq 10^5 $）——表示库苏沙的凸多边形的顶点数。接下来的 $ n $ 行按顺时针或逆时针顺序给出该多边形各顶点的坐标。第 $ i $ 行包含两个整数 $ x_i $、$ y_i $（$ |x_i|, |y_i| \leq 10^6 $）——表示按该顺序排在第 $ i $ 位的顶点的坐标。

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.

输出一个实数——所求的期望面积。

若答案的绝对误差和相对误差均不超过 $10^{-6}$，则视为正确。