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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.*;

public class Main {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();
        int[][] x = new int[n][2];
        for(int i=0; i<n; i++){
            x[i][0] = sc.nextInt();
            x[i][1] = sc.nextInt();
        }
        sc.close();

        int mod = (x[0][0] + x[0][1]) % 2;
        for(int i=1; i<n; i++){
            if((x[i][0] + x[i][1]) % 2 != mod){
                System.out.println("-1");
            }
        }

        System.out.println(40);
        int gxpos, gypos = 1, count = 1;
        if(mod == 0){
            gxpos = 1;
            System.out.print("1 ");
        }else{
            gxpos = 2;
            System.out.print("2 ");
        }
        for(int i=0; i<19; i++){
            System.out.print("1 ");
        }
        for(int i=0; i<19; i++){
            System.out.print("1 ");
        }
        System.out.println("1");
        
        for(int i = 0; i<n; i++){
            int xpos = gxpos;
            int ypos = gypos;
            count = 1;
            System.out.print("R");
            while(count < 20){
                if(x[i][0] >= xpos){
                    System.out.print("R");
                    xpos++;
                }else{
                    System.out.print("L");
                    xpos--;
                }
                count++;
            }
            if(x[i][0] > xpos){
                System.out.print("R");
                xpos++;
                count++;
            }else if(x[i][0] < xpos){
                System.out.print("L");
                xpos--;
                count++;
            }
            System.out.print("U");
            count++;
            while(count < 40){
                if(x[i][1] >= ypos){
                    System.out.print("U");
                    ypos++;
                }else{
                    System.out.print("D");
                    ypos--;
                }
                count++;
            }
            System.out.println("");
        }

        //System.out.println(20);
        // for(int i=0; i<19; i++){
        //     System.out.print("1 ");
        // }
        // System.out.println("1");
        
        // for(int i=0; i<n; i++){
        //     int count = 0;
        //     if(x[i][0] > 0){
        //         for(int j=0; j<x[i][0]; j++){
        //             System.out.print("R");
        //             count++;
        //         }
        //     }else{
        //         for(int j=0; j<-x[i][0]; j++){
        //             System.out.print("L");
        //             count++;
        //         }
        //     }
        //     if(x[i][1] > 0){
        //         for(int j=0; j<x[i][1]; j++){
        //             System.out.print("U");
        //             count++;
        //         }
        //     }else{
        //         for(int j=0; j<-x[i][1]; j++){
        //             System.out.print("D");
        //             count++;
        //         }
        //     }
        //     while(count < 20){
        //         System.out.print("U");
        //         count++;
        //     }
        //     System.out.println("");
        // }
    }
}