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: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Web;

namespace Competitive
{
    internal class Solution
    {
        public int N;
        public long MOD = 1000000007;
        public long[] X, Y;
        public int H = 36;
        
        public void Run()
        {
            N = Input.ReadInt();
            X = new long[N];
            Y = new long[N];
            for (int i = 0; i < N; i++)
            {
                var line = Input.ReadIntArray();
                X[i] = line[0];
                Y[i] = line[1];
            }

            // 偶奇が合っていない例
            long md = (X[0] + Y[0]) % 2;
            for (int i = 0; i < N; i++)
            {
                if ((X[i] + Y[i] + md) % 2 != 0)
                {
                    Console.WriteLine(-1);
                    return;
                }
            }

            // 腕の本数
            var arms = new List<long>();
            if (md == 0) arms.Add(1);
            {
                long x = 1;
                for (int i = 0; i < H; i++)
                {
                    arms.Add(x);
                    x *= 2;
                }
            }

            Console.WriteLine(arms.Count);
            Console.WriteLine(string.Join(" ", arms));
            for (int i = 0; i < N; i++)
            {
                Console.WriteLine(AnsString(X[i], Y[i]));
            }

        }

        public string AnsString(long x, long y)
        {
            string ret = "";
            if ((x + y) % 2 == 0)
            {
                ret += "R";
                x -= 1;
            }

            long u = x + y;
            long v = x - y;
            var ansu = Ans(u);
            var ansv = Ans(v);

            for (int i = 0; i < ansu.Count; i++)
            {
                int au = ansu[i];
                int av = ansv[i];
                if (au == 1 && av == 1) ret += "R";
                if (au == 1 && av == -1) ret += "U";
                if (au == -1 && av == 1) ret += "D";
                if (au == -1 && av == -1) ret += "L";
            }

            return ret;
        }

        public List<int> Ans(long a)
        {
            var ret = new List<int>();
            long x = 1;
            for (int i = 0; i < H - 1; i++)
            {
                long aa = a - x;
                if (Math.Abs(aa) % (4 * x) == 2 * x)
                {
                    a -= x;
                    ret.Add(1);
                }
                else
                {
                    a += x;
                    ret.Add(-1);
                }
                x *= 2;
            }

            if(a == x) ret.Add(1);
            else ret.Add(-1);

            return ret;
        }
    }

    // libs ----------
    

    // common ----------

    internal static class Input
    {
        public static string ReadString()
        {
            string line = Console.ReadLine();
            return line;
        }

        public static int ReadInt()
        {
            string line = Console.ReadLine();
            return int.Parse(line);
        }

        public static int[] ReadIntArray()
        {
            string line = Console.ReadLine();
            return line.Split(' ').Select(int.Parse).ToArray();            
        }

        public static long ReadLong()
        {
            string line = Console.ReadLine();
            return long.Parse(line);
        }

        public static long[] ReadLongArray()
        {
            string line = Console.ReadLine();
            return line.Split(' ').Select(long.Parse).ToArray();
        }

        public static double[] ReadDoubleArray()
        {
            string line = Console.ReadLine();
            return line.Split(' ').Select(double.Parse).ToArray();
        }
    }
    
    internal class Program
    {
        public static void Main(string[] args)
        {
            var s = new Solution();
            s.Run();
        }
    }
}