E(length(CH))

You are given $n$ points on a plane. The $i$-th point is activated with probability $p_i$. Find the expected circumference of the convex hull of the activated points.

The first line contains the number of points $n$ and the number $p^*$. Every point is activated with probability $p_i = p^*$, except the first three, which are always activated ($p_1 = p_2 = p_3 = 1$). This way, the convex hull circumference is always defined. Each of the next $n$ lines contains the coordinates $x_i$ and $y_i$ of the $i$-th point, separated by a space.

Print the expected circumference of the convex hull. Your answer is accepted if it differs from the reference solution by less than $0.001$.

$3 \le n \le 1000$; $0 \le p^* \le 1$

$0 \le x_i, y_i \le 10\,000$; $x_i, y_i \in \mathbb{Z}$

No three points are collinear. No two points share an $x$ or $y$ coordinate.