You are given $n$ points in three-dimensional space. The $i$-th point is located at $(x_i,y_i,z_i)$. Find their convex hull, and compute its number of surfaces, surface area, and volume.
The convex hull of a set of points is the smallest convex set containing all of them. A surface means a maximal planar face of its boundary. Coplanar triangles forming one face count as one surface; for example, a cube has six surfaces.
Four or more points may lie in the same plane. If all points lie in a single plane (including when all points are collinear), the hull is degenerate.
Input
The first line contains an integer $n$ ($4 \le n \le 2000$), the number of points.
Each of the next $n$ lines contains three integers $x_i$, $y_i$, and $z_i$ ($|x_i|,|y_i|,|z_i| \le 10^9$), representing the coordinates of the $i$-th point.
All points are distinct.
Output
If the hull is degenerate, output a single line containing no.
Otherwise, output yes on the first line. On the second line, output an integer $F$ and two real numbers $A$ and $V$, separated by spaces: the number of surfaces, the surface area, and the volume, respectively.
The value of $F$ must be exact. Each of $A$ and $V$ is considered correct if its absolute or relative error does not exceed $10^{-9}$. Formally, an answer $a$ and jury answer $b$ satisfy
$$|a-b| \le 10^{-9}\max(1,|b|).$$
Subtasks
| Subtask | Points | Coordinate bound |
|---|---|---|
| 1 | 25 | $|x_i|,|y_i|,|z_i| \le 10^3$ |
| 2 | 25 | $|x_i|,|y_i|,|z_i| \le 10^5$ |
| 3 | 25 | $|x_i|,|y_i|,|z_i| \le 10^7$ |
| 4 | 25 | $|x_i|,|y_i|,|z_i| \le 10^9$ |
Each test earns 100% if its entire output is correct, 40% if yes, $A$, and $V$ are correct but $F$ is incorrect, and 0% otherwise. A correct no earns 100% for that test. The score for a subtask is its point value multiplied by the minimum fraction earned over its tests.
Examples
Example 1
Input
4 0 0 0 1 0 0 0 1 0 0 0 1
Output
yes 4 2.366025403784439 0.166666666666667
Example 2
Input
8 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 1 1
Output
yes 6 6 1
Example 3
Input
4 0 0 0 1 0 1 0 1 1 1 1 2
Output
no
Notes
The first sample is a tetrahedron. The second sample is a unit cube: its six square faces count as six surfaces. In the third sample, all points lie in the plane $z=x+y$.