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: Wrong Answer)

// デバッグ実行：Ctrl+F5

using System;
using System.Collections.Generic;
class Program
{
    static void Main(string[] args)
    {
        String line;
        line = Console.ReadLine();
        int N = int.Parse(line);
        List<Tuple<int,int>> points = new List<Tuple<int,int>>();
        String[] words;
        for(int i = 0; i < N; i++)
        {
            line = Console.ReadLine();
            words = line.Split(' ');
            points.Add(new Tuple<int,int>(int.Parse(words[0]), int.Parse(words[1])));
        }
        bool Oddflag = false;
        if ((Math.Abs(points[0].Item1) + Math.Abs(points[0].Item2)) % 2 == 1)
        {
            // 奇数
            Oddflag = true;
        }
        int MaxX = 0;
        int MaxY = 0;
        foreach(Tuple<int,int> keys in points)
        {
            if (((Math.Abs(keys.Item1) + Math.Abs(keys.Item2)) % 2 == 1) != Oddflag)
            {
                Console.WriteLine("-1");
                return;
            }
            if (MaxX < Math.Abs(keys.Item1)) MaxX = Math.Abs(keys.Item1);
            if (MaxY < Math.Abs(keys.Item2)) MaxY = Math.Abs(keys.Item2);
        }
        // 答えが出せるはず
        // まずあらかじめ最大の腕を求める
        // あとは各腕に対して微調整する
        int MAX = MaxX + MaxY;
        Console.WriteLine(MAX);
        Console.Write("1");
        for (int i = 1; i < MAX; i++)
        {
            Console.Write(" 1");
        }
        Console.WriteLine();
        int Count;
        for(int i = 0; i < points.Count; i++)
        {
            Count = MAX;
            if (points[i].Item1 > 0)
            {
                for(int j = 0; j < points[i].Item1; j++)
                {
                    Console.Write("R");
                    Count--;
                }
            }
            else
            {
                for (int j = 0; j > points[i].Item1; j--)
                {
                    Console.Write("L");
                    Count--;
                }
            }
            // Y座標
            if (points[i].Item2 > 0)
            {
                for (int j = 0; j < points[i].Item2; j++)
                {
                    Console.Write("U");
                    Count--;
                }
            }
            else
            {
                for (int j = 0; j > points[i].Item2; j--)
                {
                    Console.Write("D");
                    Count--;
                }
            }
            // 余りを書き出す
            for (int j = 0; j < Count; j+=2)
            {
                Console.Write("UD");
                Count--;
            }
            Console.WriteLine();
        }
    }
}