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 (Clang 3.8.0) (judge verdict: Wrong Answer)

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>
#include <math.h>
#include <limits.h>
int max(int a, int b){return a>b?a:b;}
int min(int a, int b){return a<b?a:b;}
int gcd(int a, int b){return min(a,b)==0?max(a,b):gcd(min(a,b),max(a,b)%min(a,b));}
long long llmax(long long a, long long b){return a>b?a:b;}
long long llmin(long long a, long long b){return a<b?a:b;}
long long llgcd(long long a, long long b){return min(a,b)==0 ?llmax(a,b):llgcd(llmin(a,b),llmax(a,b)%llmin(a,b));}
int assort(const void *a, const void *b){return *(int*)a-*(int*)b;}
int dessort(const void *a, const void *b){return *(int*)b-*(int*)a;}
int llassort(const void *a, const void *b){return *(long long*)a-*(long long*)b;}
int lldessort(const void *a, const void *b){return *(long long*)b-*(long long*)a;}
int charassort(const void *a, const void *b){return strcmp((char*)a ,(char*)b);}
int chardessort(const void *a, const void *b){return strcmp((char*)b ,(char*)a);}
char lowcmp[26]="abcdefghijklmnopqrstuvwxyz";
char upcmp[26]="ABCDEFGHIJKLMNOPQRSTUVWXYZ";
int ntoi(char c){return c-'0';}
int ltoi(char c){return c-'a';}
int utoi(char c){return c-'A';}
long long mod=1000000007;

int main(void){
    int n,m=40;
    scanf("%d",&n);
    int x[n],y[n];
    for (int i=0; i<n; i++){
        scanf("%d%d",&x[i],&y[i]);
        if ( abs(x[i]+y[i])%2!=abs(x[0]+y[0])%2 ){
            printf("-1\n");
            return 0;
        }
    }
    m-=abs(x[0]+y[0])%2;
    printf("%d\n%d %d\n",m,1,1);
    for (int i=0; i<n; i++){
        
        for (int j=0; j<abs(x[i]); j++){
            if (x[i]>0){
                printf("R");
            } else {
                printf("L");
            }
        }
        for (int j=0; j<abs(y[i]); j++){
            if (y[i]>0){
                printf("U");
            } else {
                printf("D");
            }
        }        
        for (int j=0; j<(m-abs(x[i]+y[i]))/2; j++){
            printf("UD");
        }
        printf("\n");
    }
    
    
    
	return 0 ;
}