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)

using System;
using System.Collections.Generic;
class Program
{
    //static List<string> str = new List<string>();
	static void Main(string[] args)
	{
		// 入力フェーズ
        int N = int.Parse(Console.ReadLine());
        //int M = int.Parse(input[1]);
        List<joint> point = new List<joint>();

        long maxDistance = 0;
        for(int i=0; i<N; i++)
        {
            string[] input = Console.ReadLine().Split(' ');
            joint j = new joint();
            j.Xpoint = int.Parse(input[0]);
            j.Ypoint = int.Parse(input[1]);
            point.Add(j);
            long dis = Math.Abs(j.Xpoint) + Math.Abs(j.Ypoint);
            if (maxDistance < dis)
                maxDistance = dis;
        }

        // maxDistance*長さ1の腕が必要
        // 座標が(0,0)しかなかった場合、距離0だが、RLで到達できる
        if(maxDistance == 0)
        {
            Console.WriteLine("2");
            Console.WriteLine("1 1");
            for (int i = 0; i < N; i++ )
                Console.WriteLine("RL");

            return;
        }

				Console.WriteLine(maxDistance);
        // 各点について、到達できるかどうかの判定＋到達のモード列を取得する
        List<List<long>> mode = new List<List<long>>();
        for(int i=0; i<N; i++)
        {
            List<long> m = getRoute(point[i], maxDistance);
            if(m == null || m.Count == 0)
            {
                // 到達不可能
                Console.WriteLine("-1");
                return;
            }
            mode.Add(m);
        }

        /* 全て到達可能だった場合、腕がまとめられるかチェックす
        List<long> arm = new List<long>();
        // modeの第n項目を縦にまとめて昇順に差を取る
				Console.WriteLine(mode.Count + " " + mode[0].Count);
				Console.WriteLine(mode[0][0] + " " + mode[0][1] + " " + mode[0][2] + " " + mode[0][3]);
				Console.WriteLine(mode[1][0] + " " + mode[1][1] + " " + mode[1][2] + " " + mode[1][3]);
				Console.WriteLine(mode[2][0] + " " + mode[2][1] + " " + mode[2][2] + " " + mode[2][3]);
				for (int i = 0; i < N; i++ )
        {
					Console.WriteLine("in loop " + j);
            List<long> a = new List<long>(mode[i]);
            a.Sort();
						Console.WriteLine("sorted " + a[0] + " " + a[1] + " " + a[2]);

					if(i==0)
					{
						for(int j=0; j<4; j++)
            {
                if(a[j] > 0)
									arm.Add(a[j]);

									Console.WriteLine("in loop i " + arm.Count);
            }
					}
					else
					{

					}

        }
				まとめるのはとりあえず諦める
				*/

        // 腕の長さに応じてRLUDを出力する
        Console.WriteLine(maxDistance);
        for (int i = 0; i < maxDistance; i++ )
        {
						if (i > 0)
							Console.Write(" ");
            Console.Write("1");
        }
        Console.WriteLine();
        for (int i = 0; i < N; i++)
        {
            string ret = "";
						ret += new String('R', (int)mode[i][0]);
						ret += new String('L', (int)mode[i][1]);
						ret += new String('U', (int)mode[i][2]);
						ret += new String('D', (int)mode[i][3]);
            Console.WriteLine(ret);
        }

        //出力
        //Console.WriteLine(ret);
    }

    static List<long> getRoute(joint j, long m)
    {
        // 長さmの腕で、距離jの点に到達できればR,L,U,D,Aの回数を返す
        // 出来なければ""を返す
        long dis = Math.Abs(j.Xpoint) + Math.Abs(j.Ypoint);
				//bool f = (m - dis)%2 == 0;
				//Console.WriteLine(j.Xpoint + " " + j.Ypoint + " " + f);
        if((m - dis)%2 == 0)
        {
            // 到達できる
            List<long> ret = new List<long>(){0,0,0,0};
            if(j.Xpoint > 0)
            {
                ret[0] = j.Xpoint;
            }
            else
            {
                ret[1] = j.Xpoint * -1;
            }

            if(j.Ypoint > 0)
            {
                ret[2] = j.Ypoint;
            }
            else
            {
                ret[3] = j.Ypoint * -1;
            }

            if(m - dis > 0)
            {
                // R,L or U,Dの大きいほうに足す
                if(Math.Max(ret[0], ret[1]) > Math.Max(ret[2], ret[3]))
                {
                    ret[0] += (m - dis) / 2;
                    ret[1] += (m - dis) / 2;
                }
                else
                {
                    ret[2] += (m - dis) / 2;
                    ret[3] += (m - dis) / 2;
                }
            }

						return ret;
        }

        return null;
    }

    class joint
    {
        public int Xpoint { set; get; }
        public int Ypoint { set; get; }
    }

}