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.Text;
using System.Linq;
using System.Collections;
using System.Collections.Generic;
using System.IO;
using System.Numerics;
using static System.Console;
using static System.Math;

namespace AtCoder
{
    public class Program
    {
        const bool IsTest = false;

        const string t1 = "3\n-1 0\n0 3\n2 -1";
        const string t2 = "5\n0 0\n1 0\n2 0\n3 0\n4 0\n";
        const string t3 = "2\n1 1\n1 1\n";
        const string t4 = "3\n-7 -3\n7 3\n-3 -7\n";

        public static void Main(string[] args)
        {
            if (!IsTest) {
                new Program().Solve(new ConsoleInput(Console.In));
            } else {
                new Program().Solve(new ConsoleInput(new StringReader(t1)));
                new Program().Solve(new ConsoleInput(new StringReader(t2)));
                new Program().Solve(new ConsoleInput(new StringReader(t3)));
                new Program().Solve(new ConsoleInput(new StringReader(t4)));
            }
        }

        public struct Pos {
            public readonly int X;
            public readonly int Y;
            public readonly int L;
            public Pos(int x, int y)
            {
                X = x;
                Y = y;
                L = (x > 0 ? x : -x) + (y > 0 ? y : -y);
            }
        }

        public void Solve(ConsoleInput cin)
        {
            cin.ReadLine();
            var n = cin.Read<int> ();
            var v = new List<Pos> ();
            var max = 0;
            // 部分点を取りに行く
            for (var i = 0; i < n; i++) {
                var t = cin.ReadLineAs<int, int> ();
                var p = new Pos(t.Item1, t.Item2); 
                v.Add(p);
                if (p.L > max) {
                    max = p.L;
                }
            }
            for (var i = 0; i < n; i++) {
                if (((max - v[i].L) % 2) != 0) {
                    Console.WriteLine(-1);
                    return;
                }
            }
            if (max > 40) {
                // 無駄にCPUつかうのもなんなのであきらめる
                Console.WriteLine(-1);
                return;
            }
            Console.WriteLine(max);
            var c = "1";
            for (var i = 1; i < max; i++) {
                c += " 1";
            }
            Console.WriteLine(c);

            for (var i = 0; i < n; i++) {
                c = "";
                for (var j = 0; j < max - v[i].L; j += 2) {
                    c += "UD";
                }
                if (v[i].X > 0) {
                    for (var j = 0; j < v[i].X; j++) {
                        c += "R";
                    }
                } else if (v[i].X < 0) {
                    for (var j = 0; j < -v[i].X; j++) {
                        c += "L";
                    }
                }
                if (v[i].Y > 0) {
                    for (var j = 0; j < v[i].Y; j++) {
                        c += "U";
                    }
                } else if (v[i].Y < 0) {
                    for (var j = 0; j < -v[i].Y; j++) {
                        c += "D";
                    }
                }
                Console.WriteLine(c);
            }
        }
    }

    public class ConsoleInput
    {
        private readonly System.IO.TextReader _stream;
        private readonly char _separator = ' ';
        private string _curLine;
        private string[] _curLineSplitted;
        private int _curItemIndex;
        public ConsoleInput(System.IO.TextReader stream, char separator = ' ')
        {
            this._separator = separator;
            this._stream = stream;
        }
        public T Read<T>()
        {
            var converter = System.ComponentModel.TypeDescriptor.GetConverter(typeof(T));
            if (converter == null) {
                throw new Exception(string.Format("TypeDescriptor.GetConverter {0}", typeof(T).Name));
            }
            return (T)converter.ConvertFromString(ReadString());
        }
        public T[] ReadArray<T>(long N)
        {
            var ret = new T[N];
            for (long i = 0; i < N; i++) {
                ret[i] = Read<T>();
            }
            return ret;
        }
        public Tuple<T> ReadLineAs<T>()
        {
            ReadLine();
            return new Tuple<T>(Read<T>());
        }
        public Tuple<T, S> ReadLineAs<T, S>()
        {
            ReadLine();
            return new Tuple<T, S>(Read<T>(), Read<S>());
        }
        public Tuple<T, S, U> ReadLineAs<T, S, U>()
        {
            ReadLine();
            return new Tuple<T, S, U>(Read<T>(), Read<S>(), Read<U>());
        }
        public Tuple<T, S, U, V> ReadLineAs<T, S, U, V>()
        {
            ReadLine();
            return new Tuple<T, S, U, V>(Read<T>(), Read<S>(), Read<U>(), Read<V>());
        }
        public Tuple<T, S, U, V, R> ReadLineAs<T, S, U, V, R>()
        {
            ReadLine();
            return new Tuple<T, S, U, V, R>(Read<T>(), Read<S>(), Read<U>(), Read<V>(), Read<R>());
        }

        public string ReadString()
        {
            return _curLineSplitted[_curItemIndex++];
        }
        public void ReadLine()
        {
            _curLine = _stream.ReadLine();
            if (_curLine == null) {
                throw new System.Exception("No more lines");
            }
            _curLineSplitted = _curLine.Split(_separator);
            _curItemIndex = 0;
        }
    }
}