Let $S$ be a sphere with radius $1$ and center $(0, 0, 0)$. Let $a_0, a_1, \\dots, a_n$ be $n + 1$ points on the surface of $S$. The positions of $a_1, \\dots, a_n$ 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 $a_0, \\dots, a_n$(possibly on the border) and $0$ otherwise. Calculate the expected value of $f$.
The first line contains an integer $n$ denoting the number of points ($0 <= n <= 100000$).
The $i$-th line of the next $n$ lines contains three integers $x, y, z$ denoting the point $a_i = (frac(x, sqrt(x^2 + y^2 + z^2)), frac(y, sqrt(x^2 + y^2 + z^2)), frac(z, sqrt(x^2 + y^2 + z^2)))$ ($-1000000 <= x, y, z <= 1000000, x^2 + y^2 + z^2 eq.not 0$).
It is guaranteed that $a_1, \\dots, a_n$ are distinct.
Output the answer.
The answer will be considered correct if its absolute or relative error doesn't exceed $10^(-6)$.
## Input
The first line contains an integer $n$ denoting the number of points ($0 <= n <= 100000$).The $i$-th line of the next $n$ lines contains three integers $x, y, z$ denoting the point $a_i = (frac(x, sqrt(x^2 + y^2 + z^2)), frac(y, sqrt(x^2 + y^2 + z^2)), frac(z, sqrt(x^2 + y^2 + z^2)))$ ($-1000000 <= x, y, z <= 1000000, x^2 + y^2 + z^2 eq.not 0$).It is guaranteed that $a_1, \\dots, a_n$ are distinct.
## Output
Output the answer.The answer will be considered correct if its absolute or relative error doesn't exceed $10^(-6)$.
[samples]
**Definitions**
Let $ S = \{ \mathbf{x} \in \mathbb{R}^3 : \|\mathbf{x}\| = 1 \} $ be the unit sphere centered at the origin.
Let $ \mathbf{a}_1, \dots, \mathbf{a}_n \in S $ be fixed distinct points on $ S $.
Let $ \mathbf{a}_0 $ be a random point uniformly distributed on $ S $.
Define the indicator function:
$$
f = \begin{cases}
1 & \text{if } \exists \text{ a closed hemisphere } H \subseteq \mathbb{R}^3 \text{ such that } \{\mathbf{a}_0, \dots, \mathbf{a}_n\} \subseteq H, \\
0 & \text{otherwise}.
\end{cases}
$$
**Constraints**
1. $ n \in \mathbb{Z} $, $ 0 \leq n \leq 100000 $
2. For each $ i \in \{1, \dots, n\} $, $ \mathbf{a}_i = \frac{(x_i, y_i, z_i)}{\| (x_i, y_i, z_i) \|} $ for integers $ x_i, y_i, z_i \in [-10^6, 10^6] $, with $ x_i^2 + y_i^2 + z_i^2 \neq 0 $.
3. All $ \mathbf{a}_1, \dots, \mathbf{a}_n $ are distinct.
**Objective**
Compute $ \mathbb{E}[f] $, the probability that $ \mathbf{a}_0, \mathbf{a}_1, \dots, \mathbf{a}_n $ lie together in some closed hemisphere.