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;
using System.Threading.Tasks;
using System.Diagnostics;

public class Program
{
    public static readonly int Mod = 1000000007;

    public static void Main(string[] args)
    {
        int N = Scanner.Int;

        var X = new long[N];
        var Y = new long[N];
        foreach (int i in Enumerable.Range(0, N))
        {
            X[i] = Scanner.Int;
            Y[i] = Scanner.Int;
        }

        var V = Enumerable.Range(0, 32).Select(i => 1L << i).ToList();

        var mods = X.Zip(Y, (x, y) => Math.Abs(x) + Math.Abs(y));

        bool AllOdd = mods.Where(i => i % 2 == 1).Count() == N;
        bool AllEven = mods.Where(i => i % 2 == 0).Count() == N;

        //  無理
        if (!AllEven && !AllOdd)
        {
            Console.WriteLine(-1);
            return;
        }

        //  偶数ならば1を追加
        if (AllEven)
        {
            V.Insert(0, 1);
        }

        Console.WriteLine(V.Count());
        Console.WriteLine(string.Join(" ", V));

        foreach (int i in Enumerable.Range(0, N))
        {
            string ans = "";
            foreach (int j in Enumerable.Range(0, V.Count()).Reverse())
            {
                if (Math.Abs(X[i]) >= Math.Abs(Y[i]))
                {
                    ans += (X[i] >= 0) ? 'L' : 'R';
                    X[i] += (X[i] >= 0) ? -V[j] : V[j];
                }
                else
                {
                    ans += (Y[i] >= 0) ? 'D' : 'U';
                    Y[i] += (Y[i] >= 0) ? -V[j] : V[j];
                }
            }
            Console.WriteLine(ans);
        }

        Console.WriteLine();
        Console.ReadLine();
    }
}

/// <summary>
/// スキャナ
/// </summary>
public static class Scanner
{
    public static char[] separator = new char[] { ' ' };

    private static IEnumerator<string> enumerator = Enumerator();
    private static IEnumerator<string> Enumerator()
    {
        while (true)
        {
            foreach (string s in Console.ReadLine().Split(separator))
            {
                yield return s;
            }
        }
    }

    public static IEnumerable<string> StringLine
    {
        get
        {
            return Console.ReadLine().Split(separator);
        }
    }
    public static IEnumerable<int> IntLine
    {
        get
        {
            return StringLine.Select(int.Parse);
        }
    }
    public static IEnumerable<long> LongLine
    {
        get
        {
            return StringLine.Select(long.Parse);
        }
    }
    public static IEnumerable<double> DoubleLine
    {
        get
        {
            return StringLine.Select(double.Parse);
        }
    }
    public static IEnumerable<char> CharLine
    {
        get
        {
            return StringLine.Select(char.Parse);
        }
    }

    public static string String
    {
        get
        {
            return enumerator.Next();
        }
    }
    public static int Int
    {
        get
        {
            return String.Int();
        }
    }
    public static long Long
    {
        get
        {
            return String.Long();
        }
    }
    public static double Double
    {
        get
        {
            return String.Double();
        }
    }
    public static char Char
    {
        get
        {
            return char.Parse(String);
        }
    }

    public static Tuple<int, int> IntTuple
    {
        get
        {
            return new Tuple<int, int>(Int, Int);
        }
    }

}

/// <summary>
/// Linq拡張
/// </summary>
public static class LinqExtension
{
    /// <summary>
    /// 二分探索
    /// </summary>
    /// <typeparam name="T"></typeparam>
    /// <param name="self"></param>
    /// <param name="solve"></param>
    /// <param name="min"></param>
    /// <param name="max"></param>
    /// <returns></returns>
    public static T LeftBinarySearch<T>(this IEnumerable<T> self, Func<T, bool> solve, T min, T max)
    {
        int index = self.LeftBinarySearchIndex(solve);

        return self.Get(index, min, max);
    }

    /// <summary>
    /// 二分探索
    /// </summary>
    /// <typeparam name="T"></typeparam>
    /// <param name="self"></param>
    /// <param name="solve"></param>
    /// <param name="min"></param>
    /// <param name="max"></param>
    /// <returns></returns>
    public static T RightBinarySearch<T>(this IEnumerable<T> self, Func<T, bool> solve, T min, T max)
    {
        int index = self.RightBinarySearchIndex(solve);

        return self.Get(index, min, max);
    }

    /// <summary>
    /// <para>二分探索</para>
    /// <para>渡されたラムダ式の条件を満たすものの中で、一番左側のもの(インデックスが小さい)</para>
    /// </summary>
    /// <typeparam name="T">IEnumerable</typeparam>
    /// <param name="solve"></param>
    /// <returns>一番左側のインデックス</returns>
    public static int LeftBinarySearchIndex<T>(this IEnumerable<T> self, Func<T, bool> solve)
    {
        int ng = -1;
        int ok = self.Count();

        while (Math.Abs(ok - ng) > 1)
        {
            int mid = (ok + ng) / 2;

            if (solve(self.ElementAt(mid)))
            {
                ok = mid;
            }
            else
            {
                ng = mid;
            }
        }

        return ok;
    }

    /// <summary>
    /// <para>二分探索</para>
    /// <para>渡されたラムダ式の条件を満たすものの中で、一番右側のもの(インデックスが大きい)</para>
    /// </summary>
    /// <typeparam name="T">IEnumerable</typeparam>
    /// <param name="solve"></param>
    /// <returns>一番右側のインデックス</returns>
    public static int RightBinarySearchIndex<T>(this IEnumerable<T> self, Func<T, bool> solve)
    {
        int ng = self.Count();
        int ok = -1;

        while (Math.Abs(ok - ng) > 1)
        {
            int mid = (ok + ng) / 2;

            if (solve(self.ElementAt(mid)))
            {
                ok = mid;
            }
            else
            {
                ng = mid;
            }
        }

        return ok;
    }

    /// <summary>
    /// OutOfRange対応
    /// </summary>
    /// <typeparam name="T">型T</typeparam>
    /// <param name="self">IEnumerable</param>
    /// <param name="index">対象のインデックス</param>
    /// <param name="min">0未満のインデックスの時</param>
    /// <param name="max">配列のサイズ以上の時</param>
    /// <returns>型Tの値</returns>
    public static T Get<T>(this IEnumerable<T> self, int index, T min, T max)
    {
        if (index < 0) return min;
        if (index >= self.Count()) return max;

        return self.ElementAt(index);
    }

    /// <summary>
    /// リストを埋める
    /// </summary>
    /// <typeparam name="T">型T</typeparam>
    /// <param name="self">リスト</param>
    /// <param name="func">(index) => T</param>
    /// <returns>リストの参照</returns>
    public static IList<T> Fill<T>(this IList<T> self, Func<int, T> func)
    {
        for (int i = 0; i < self.Count; i++)
        {
            self[i] = func(i);
        }

        return self;
    }

    /// <summary>
    /// リストを埋める
    /// </summary>
    /// <typeparam name="T">型T</typeparam>
    /// <param name="self">リスト</param>
    /// <param name="value">型Tの値</param>
    /// <returns>リストの参照</returns>
    public static IList<T> Fill<T>(this IList<T> self, T value)
    {
        for (int i = 0; i < self.Count; i++)
        {
            self[i] = value;
        }

        return self;
    }

    public static T Next<T>(this IEnumerator<T> self)
    {
        self.MoveNext();
        return self.Current;
    }

    public static void Write(this IEnumerable<bool> self)
    {
        StringBuilder sb = new StringBuilder();
        foreach(bool b in self)
        {
            sb.Append(b ? "□" : "■");
        }
        Console.WriteLine(sb.ToString());
    }
}

/// <summary>
/// Int拡張
/// </summary>
public static class IntExtensiton
{
    /// <summary>
    /// 階上
    /// </summary>
    /// <param name="self"></param>
    /// <returns></returns>
    public static long Factorial(this int self)
    {
        if (self == 1) return 1L;
        return self * (self - 1).Factorial();
    }

    /// <summary>
    /// 階上
    /// </summary>
    /// <param name="self"></param>
    /// <returns></returns>
    public static long Factorial(this int self, int c)
    {
        if (c == 0 || self == 1) return 1L;
        return self * Factorial(self - 1, c - 1);
    }

    public static int BitCount(this int bit, int digit = 32)
    {
        int count = 0;
        for (int i = (digit - 1); i >= 0; i--)
        {
            if (bit.GetBit(i)) count++;
        }
        return count;
    }

    public static bool GetBit(this int bit, int digit)
    {
        int mask = (1 << digit);
        return (bit & mask) == mask;
    }

    public static int SetBit(this int bit, int digit)
    {
        return (bit | (1 << digit));
    }

    public static string BitToString(this int bit, int digit = 32)
    {
        StringBuilder sb = new StringBuilder();

        for (int i = (digit - 1); i >= 0; i--)
        {
            sb.Append(bit.GetBit(i) ? "1" : "0");
        }

        return sb.ToString();
    }

    public static bool IsDivisor(this int self, int target)
    {
        return self % target == 0;
    }

    public static IEnumerable<int> Divisors(this int self)
    {
        for (int i = 1; i * i <= self; i++)
        {
            if (self % i == 0)
            {
                yield return i;
                if (i * i != self) yield return self / i;
            }
        }
    }

    public static bool NeighboringBit(this int self, int digit = 32)
    {
        if (digit < 2) return false;
        return Enumerable.Range(0, digit - 1).Where(j => self.GetBit(j) && self.GetBit(j + 1)).Any();
    }
}

/// <summary>
/// Long拡張
/// </summary>
public static class LongExtensition
{
    public static long BitCount(this long bit, int digit = 64)
    {
        long count = 0;
        for (int i = (digit - 1); i >= 0; i--)
        {
            if (bit.GetBit(i)) count++;
        }
        return count;
    }

    public static bool GetBit(this long bit, int digit)
    {
        long mask = (1L << digit);
        return (bit & mask) == mask;
    }

    public static long SetBit(this long bit, int digit)
    {
        return bit | (1L << digit);
    }

    public static string BitToString(this long bit, int digit = 64)
    {
        StringBuilder sb = new StringBuilder();

        for (int i = (digit - 1); i >= 0; i--)
        {
            sb.Append(bit.GetBit(i) ? "1" : "0");
        }

        return sb.ToString();
    }

    public static bool IsDivisor(this long self, int target)
    {
        return self % target == 0;
    }
}

/// <summary>
/// String拡張
/// </summary>
public static class StringExtensiton
{
    public static void Print(this string str) => Console.WriteLine(str);
    public static int Int(this string str) => int.Parse(str);
    public static long Long(this string str) => long.Parse(str);
    public static double Double(this string str) => double.Parse(str);
}

/// <summary>
/// 便利関数
/// </summary>
public static class F
{
    public static void ChMax<T>(ref T x, T y) where T : IComparable
    {
        if (x.CompareTo(y) < 0)
        {
            x = y;
        }
    }

    public static void ChMin<T>(ref T x, T y) where T : IComparable
    {
        if (x.CompareTo(y) > 0)
        {
            x = y;
        }
    }

    public static void Swap<T>(ref T x, ref T y)
    {
        T tmp = x;

        x = y;
        y = tmp;
    }
}