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;

class Program
{
    static void Main(string[] args)
    {
        long n = long.Parse(System.Console.ReadLine().Trim());
        long[][] xy = new long[n][]; 

        for(int i = 0; i < n; i++)
        {
            xy[i] = Array.ConvertAll(System.Console.ReadLine().Trim().Split(' '), long.Parse);
        }

        //全ての偶奇が同一でない場合はNG
        long parity = Math.Abs(xy[0][0] + xy[0][1]) % 2;
        for (int i = 1; i < xy.Length; i++)
        {
            if(Math.Abs(xy[i][0] + xy[i][1]) % 2 != parity)
            {

                System.Console.WriteLine("{0}",-1);
                return;
            }
        }

        //最も遠い地点の距離
        long farP = 0;
        for (int i = 0; i < xy.Length; i++)
        {
            farP = Math.Max(farP, Math.Abs(xy[i][0]) + Math.Abs(xy[i][1]));
        }

        //最長のアーム
        long d1 = (long)Math.Pow(2, (long)(Math.Log(farP - 1, 2)));

        //アームコレクション
        List<long> d = new List<long>();
        for(long i = d1;i > 0;i /= 2)
        {
            d.Add(i);
        }
        if (parity == 0) { d.Add(1); }//偶数の場合は腕を追加


        System.Console.WriteLine("{0}", d.Count);
        System.Console.WriteLine("{0}", string.Join(" ", Array.ConvertAll(d.ToArray(), i => i.ToString())));

        long x ,y;
        long distL, distR, distU, distD;
        string rlt = string.Empty;



        //出力のパターンを表示
        for (int i = 0; i < xy.Length; i++)
        {

            x = 0; y = 0;
            distL = 0; distR = 0; distU = 0; distD = 0;
            rlt = string.Empty;
  
            for (int j = 0; j < d.Count; j++)
            {
                distR = (long)Math.Pow(Math.Abs(xy[i][0] - (x + d[j])),2) + (long)Math.Pow(Math.Abs(xy[i][1] - y), 2);
                distL = (long)Math.Pow(Math.Abs(xy[i][0] - (x - d[j])), 2) + (long)Math.Pow(Math.Abs(xy[i][1] - y), 2);
                distU = (long)Math.Pow(Math.Abs(xy[i][0] - x ), 2) + (long)Math.Pow(Math.Abs(xy[i][1] - (y + d[j])), 2);
                distD = (long)Math.Pow(Math.Abs(xy[i][0] - x ), 2) + (long)Math.Pow(Math.Abs(xy[i][1] - (y - d[j])), 2);

                if (distR <= distL && distR <= distU && distR <= distD)
                {
                    rlt += "R";
                    x += d[j];
                }
                if (distL <= distR && distL <= distU && distL <= distD)
                {
                    rlt += "L";
                    x -= d[j];
                }
                if (distU <= distR && distU <= distL && distU <= distD)
                {
                    rlt += "U";
                    y += d[j];
                }
                if (distD <= distR && distD <= distL && distD <= distU)
                {
                    rlt += "D";
                    y -= d[j];
                }
            }

            System.Console.WriteLine("{0}", rlt);
        }
        Console.ReadKey(true);

    }
}