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;
using System.Linq;
using System.Text;

namespace Practice
{
    class Program
    {
        private static int ReadAndParseInt()
        {
            return int.Parse(Console.ReadLine());
        }
        private static int[] ReadAndParseIntArr()
        {
            return Array.ConvertAll(Console.ReadLine().Split(' '), int.Parse);
        }
        private static long ReadAndParseLong()
        {
            return long.Parse(Console.ReadLine());
        }
        private static long[] ReadAndParseLongArr()
        {
            return Array.ConvertAll(Console.ReadLine().Split(' '), long.Parse);
        }
        static void Main(string[] args)
        {
            SolveA(args);
            //SolveB(args);
            //SolveC(args);
            //SolveD(args);


        }
        static void SolveA(string[] args)
        {
            int n = ReadAndParseInt();
            long[] X = new long[n];
            long[] Y = new long[n];
            for(int i=0;i<n;++i)
            {
                var xy = ReadAndParseIntArr();
                X[i] = xy[0];
                Y[i] = xy[1];
            }

            if(!IsValid(X, Y))
            {
                Console.WriteLine(-1);
                return;
            }

            long maxLen = FindMaxLen(X, Y);
            // 1マスずつ進むぞ
            Console.WriteLine(maxLen);
            Console.WriteLine(GetDistanceStr(maxLen));
            for(int i=0;i<n;++i)
            {
                Console.WriteLine(GetW(maxLen, X[i], Y[i]));
            }
        }
       
        private static bool IsValid(long[] x, long [] y)
        {
            long lastMod = (x[0] + y[0]) % 2;
            for(int i=1;i<x.Length;++i)
            {
                if( (x[i] + y[i]) % 2 != lastMod)
                {
                    return false;
                }

            }
            return true;
        }
        private static long FindMaxLen(long[] x, long[] y)
        {
            long max = 0;
            for(int i=0;i<x.Length;++i)
            {
                long len = Math.Abs(x[i]) + Math.Abs(y[i]);
                if( len > max)
                {
                    max = len;
                }
            }
            return max;
        }
        private static string GetDistanceStr(long m)
        {
            string sz = "";
            for(int i=0;i<m;++i)
            {
                sz += "1 ";
            }
            return sz;
        }
        private static string GetW(long m, long x, long y)
        {
            string sz = "";
            // x方向
            long len = Math.Abs(x);
            for(int i=0;i<len;++i)
            {
                sz += GetModeX(x);
                m--;
            }

            // y方向
            len = Math.Abs(y);
            for(int i=0;i<len;++i)
            {
                sz += GetModeY(y);
                m--;
            }

            // 残った回数x方向に行ったり来たり
            for(int i=0;i<m;++i)
            {
                if(i % 2 == 0)
                {
                    sz += 'R';
                }
                else
                {
                    sz += 'L';
                }
            }
            return sz;
        }
        private static char GetModeX(long x)
        {
            if( x > 0) { return 'R'; }
            else { return 'L'; }
        }
        private static char GetModeY(long y)
        {
            if( y > 0) { return 'U'; }
            else { return 'D'; }
        }
    }
}