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++14 (GCC 5.4.1) (judge verdict: Runtime Error)

//jinhaonan
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef double db;
template<class T>inline void read(T &x){
	x=0;char c=getchar();bool f=0;
	for(;c>'9'||c<'0';f=c=='-',c=getchar());
	for(;c>='0'&&c<='9';x=(x<<1)+(x<<3)+c-'0',c=getchar());
	x=f?-x:x;
}
template<class T>inline void write(T x){
	if(x<0)putchar('-'),x=-x;
	if(x>=10)write(x/10);
	putchar(x%10+48);
}
template<class T>inline bool chkmax(T &a,T const &b){
	return a<b?a=b,1:0;
}
template<class T>inline bool chkmin(T &a,T const &b){
	return a>b?a=b,1:0;
}
const int N=1000+10;
int x[N],y[N],d[32];
void Main(){
	int n;read(n);
	bool flag[2]={false,false};
	for(int i=1;i<=n;++i){
		read(x[i]),read(y[i]);
		flag[(x[i]+y[i])&1]=1;
	}
	if(flag[0]&&flag[1]){
		puts("-1");
		return;
	}
	int m=30;
	for(int i=0;i<m;++i){
		d[i]=1<<i;
	}
	reverse(d,d+m);
	if(flag[0]){
		d[m]=1;
		write(m+1),putchar('\n');
		for(int i=0;i<=m;++i){
			write(d[i]),putchar(i==m?'\n':' ');
		}
	}else{
		write(m),putchar('\n');
		for(int i=0;i<m;++i){
			write(d[i]),putchar(i==m-1?'\n':' ');
		}
	}
	for(int i=1;i<=n;++i){
		if(flag[0]){
			--x[i];
		}
		for(int j=0;j<m;++j){
			bool u=y[i]+x[i]>0,v=y[i]-x[i]>0;
			if(u&&v){
				putchar('U');
				y[i]-=d[j];
			}else if(!u&&!v){
				putchar('D');
				y[i]+=d[j];
			}else if(u&&!v){
				putchar('R');
				x[i]-=d[j];
			}else if(!u&&v){
				putchar('L');
				x[i]+=d[j];
			}
		}
		assert(x[i]==0&&y[i]==0);
		if(flag[0]){
			putchar('R');
		}
		putchar('\n');
	}
}
int main(){
    // freopen("in.txt","r",stdin);
    // int test;read(test);
    // while(test--){
        Main();
    // }
    return 0;
}