Problem:
Score : 600 points

Problem Statement

Snuke is introducing a robot arm with the following properties to his factory:

The robot arm consists of m sections and m+1 joints. The sections are numbered 1, 2, ..., m, and the joints are numbered 0, 1, ..., m. Section i connects Joint i-1 and Joint i. The length of Section i is d_i.

For each section, its mode can be specified individually. There are four modes: L, R, D and U. The mode of a section decides the direction of that section. If we consider the factory as a coordinate plane, the position of Joint i will be determined as follows (we denote its coordinates as (x_i, y_i)):

(x_0, y_0) = (0, 0).

If the mode of Section i is L, (x_{i}, y_{i}) = (x_{i-1} - d_{i}, y_{i-1}).

If the mode of Section i is R, (x_{i}, y_{i}) = (x_{i-1} + d_{i}, y_{i-1}).

If the mode of Section i is D, (x_{i}, y_{i}) = (x_{i-1}, y_{i-1} - d_{i}).

If the mode of Section i is U, (x_{i}, y_{i}) = (x_{i-1}, y_{i-1} + d_{i}).

Snuke would like to introduce a robot arm so that the position of Joint m can be matched with all of the N points (X_1, Y_1), (X_2, Y_2), ..., (X_N, Y_N) by properly specifying the modes of the sections.
Is this possible?
If so, find such a robot arm and how to bring Joint m to each point (X_j, Y_j).

Constraints

All values in input are integers.

1 \leq N \leq 1000

-10^9 \leq X_i \leq 10^9

-10^9 \leq Y_i \leq 10^9

Partial Score

In the test cases worth 300 points, -10 \leq X_i \leq 10 and -10 \leq Y_i \leq 10 hold.

Input

Input is given from Standard Input in the following format:

N
X_1 Y_1
X_2 Y_2
:
X_N Y_N

Output

If the condition can be satisfied, follow the following format. If the condition cannot be satisfied, print -1.

m
d_1 d_2 ... d_m
w_1
w_2
:
w_N

m and d_i are the configurations of the robot arm. Refer to the problem statement for what each of them means.
Here, 1 \leq m \leq 40 and 1 \leq d_i \leq 10^{12} must hold. Also, m and d_i must all be integers.

w_j is a string of length m that represents the way to bring Joint m of the robot arm to point (X_j, Y_j).
The i-th character of w_j should be one of the letters L, R, D and U, representing the mode of Section i.

Sample Input 1

3
-1 0
0 3
2 -1

Sample Output 1

2
1 2
RL
UU
DR

In the given way to bring Joint m of the robot arm to each (X_j, Y_j), the positions of the joints will be as follows:

To (X_1, Y_1) = (-1, 0): First, the position of Joint 0 is (x_0, y_0) = (0, 0). As the mode of Section 1 is R, the position of Joint 1 is (x_1, y_1) = (1, 0). Then, as the mode of Section 2 is L, the position of Joint 2 is (x_2, y_2) = (-1, 0).

To (X_2, Y_2) = (0, 3): (x_0, y_0) = (0, 0), (x_1, y_1) = (0, 1), (x_2, y_2) = (0, 3).

To (X_3, Y_3) = (2, -1): (x_0, y_0) = (0, 0), (x_1, y_1) = (0, -1), (x_2, y_2) = (2, -1).

Sample Input 2

5
0 0
1 0
2 0
3 0
4 0

Sample Output 2

-1

Sample Input 3

2
1 1
1 1

Sample Output 3

2
1 1
RU
UR

There may be duplicated points among (X_j, Y_j).

Sample Input 4

3
-7 -3
7 3
-3 -7

Sample Output 4

5
3 1 4 1 5
LRDUL
RDULR
DULRD

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using static System.Console;
using static System.Int32;
using static System.Math;
 
class Program
{
    public struct Point
    {
        public int X;
        public int Y;
    }
    static void Main(string[] args)
    {
        label:
        int n = Parse(ReadLine());
        var points = new Point[n];
        int gusuKisuFail = 0;
        int maxDiff = 0;
        for (int i = 0; i < points.Length; ++i)
        {
            var sp = ReadLine().Split();
            points[i].X = Parse(sp[0]);
            points[i].Y = Parse(sp[1]);
            var tmp = Abs(points[i].X) + Abs(points[i].Y);
            if (i == 0) gusuKisuFail = tmp % 2;
            else if (gusuKisuFail != tmp % 2) gusuKisuFail = 2;
            maxDiff = Max(maxDiff, tmp);
        }
        if (gusuKisuFail == 2)
        {
            WriteLine("-1");
        }
        else
        {
            maxDiff /= 2;
            int tmp = 1;
            while (tmp < maxDiff) tmp *= 2;
            var arms = new List<int>();
            while (0 < tmp)
            {
                arms.Add(tmp);
                tmp /= 2;
            }
            arms.Add(1);
            arms.Add(1);
            arms.Add(1);
            if (gusuKisuFail == 1) arms.Add(1);
            WriteLine(arms.Count);
            WriteLine(string.Join(" ", arms.Select(a => a.ToString())));
            for (int j = 0; j < points.Length; ++j)
            {
                var p = points[j];
                var array = new char[arms.Count];
                for (int i = 0; i < arms.Count; ++i)
                {
                    if (Abs(p.X) < Abs(p.Y))
                    {
                        if (0 < p.Y)
                        {
                            array[i] = 'U';
                            p.Y -= arms[i];
                        }
                        else
                        {
                            array[i] = 'D';
                            p.Y += arms[i];
                        }
                    }
                    else
                    {
                        if (0 < p.X)
                        {
                            array[i] = 'R';
                            p.X -= arms[i];
                        }
                        else
                        {
                            array[i] = 'L';
                            p.X += arms[i];
                        }

                    }
                }
                WriteLine(new string(array));
                //WriteLine($"({p.X},{p.Y}");
            }
        }
        //goto label;
    }
}