E(length(CH))

You are given $n$ points on a plane. The i-th of them is activated with probability $p\_i$, provided as part of the input. Find the expected circumference of the convex hull of all activated points.

The first line contains the number of points $n$ and the number $p^∗$. All points are activated with the probability $p\_i = p^∗$, except for the first three, which are always activated ($p\_1 = p\_2 = p\_3 = 1$); this way, the definition of the convex hull circumference is always sound. This is followed by $n$ lines, each containing space-separated coordinates of the $i$-th point, $x\_i$, $y\_i$.

Print out a single number — the expected circumference of the convex hull. Your result should differ by less than $0.001$ from the reference solution to be considered correct.

• $3 ≤ n ≤ 1000$; $0 ≤ p^∗ ≤ 1$
• $0 ≤ x\_i , y\_i ≤ 10\\,000$; $x\_i , y\_i ∈ Z$
• No three points are colinear. No two points share an $x$ or $y$ coordinate.