23459번 - Moon 스페셜 저지다국어

문제

Let $S$ be a sphere with radius 1ドル$ 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ドル$ if there exists a hemisphere of $S$ that contains $a_0,\ldots,a_n$ and 0ドル$ otherwise. Calculate the expected value of $f$.

입력

The first line contains an integer $n$ denoting the number of points (0ドル\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.

출력

Output the answer.

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

제한

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