P13946 [EC Final 2019] Moon

Let $S$ be a sphere with radius $1$ and center $(0, 0, 0)$. Let $a_0,a_1,\ldots,a_n$ be $n+1$ points on the surface of $S$. The positions of $a_1,\ldots,a_n$ are fixed while the position of $a_0$ is a uniform random point on the surface of $S$. Let $f$ be $1$ if there exists a hemisphere of $S$ that contains $a_0,\ldots,a_n$ and $0$ otherwise. Calculate the expected value of $f$.

The first line contains an integer $n$ denoting the number of points ($0\le n\le 100000$). The $i$-th line of the next $n$ lines contains three integers $x, y, z$ denoting the point $a_i=\left(\frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}}\right)$ ($-1000000\le x, y, z\le 1000000, x^2+y^2+z^2\neq 0$). It is guaranteed that $a_1,\ldots,a_n$ are distinct.

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