Cubic Lattice

4000ms

262144K

Description:

A cubic lattice $$$L$$$ in $$$3$$$-dimensional euclidean space is a set of points defined in the following way: $$$$$$L=\{u \cdot \vec r_1 + v \cdot \vec r_2 + w \cdot \vec r_3\}_{u, v, w \in \mathbb Z}$$$$$$ Where $$$\vec r_1, \vec r_2, \vec r_3 \in \mathbb{Z}^3$$$ are some integer vectors such that:

$$$\vec r_1$$$, $$$\vec r_2$$$ and $$$\vec r_3$$$ are pairwise orthogonal: $$$$$$\vec r_1 \cdot \vec r_2 = \vec r_1 \cdot \vec r_3 = \vec r_2 \cdot \vec r_3 = 0$$$$$$ Where $$$\vec a \cdot \vec b$$$ is a dot product of vectors $$$\vec a$$$ and $$$\vec b$$$.
$$$\vec r_1$$$, $$$\vec r_2$$$ and $$$\vec r_3$$$ all have the same length: $$$$$$|\vec r_1| = |\vec r_2| = |\vec r_3| = r$$$$$$

You're given a set $$$A=\{\vec a_1, \vec a_2, \dots, \vec a_n\}$$$ of integer points, $$$i$$$-th point has coordinates $$$a_i=(x_i;y_i;z_i)$$$. Let $$$g_i=\gcd(x_i,y_i,z_i)$$$. It is guaranteed that $$$\gcd(g_1,g_2,\dots,g_n)=1$$$.

You have to find a cubic lattice $$$L$$$ such that $$$A \subset L$$$ and $$$r$$$ is the maximum possible.

Input:

First line contains single integer $$$n$$$ ($$$1 \leq n \leq 10^4$$$) — the number of points in $$$A$$$.

The $$$i$$$-th of the following $$$n$$$ lines contains integers $$$x_i$$$, $$$y_i$$$, $$$z_i$$$ ($$$0 < x_i^2 + y_i^2 + z_i^2 \leq 10^{16}$$$) — coordinates of the $$$i$$$-th point.

It is guaranteed that $$$\gcd(g_1,g_2,\dots,g_n)=1$$$ where $$$g_i=\gcd(x_i,y_i,z_i)$$$.

Output:

In first line output a single integer $$$r^2$$$, the square of maximum possible $$$r$$$.

In following $$$3$$$ lines output coordinates of vectors $$$\vec r_1$$$, $$$\vec r_2$$$ and $$$\vec r_3$$$ respectively.

If there are multiple possible answers, output any.

Sample Input:

2
1 2 3
1 2 1

Sample Output:

Sample Input:

1
1 2 2

Sample Output:

Sample Input:

1
2 5 5

Sample Output:

Informação

Provedor Codeforces

Código CF1375I