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: Wrong Answer)

#include <iostream>
#include <string>
#include <algorithm>
#include <vector>
#include <set>
#include <queue>
#include <stack>
#include <map>
#include <functional>
#include <math.h>
using namespace std;

#define ll long long
//#define int long long
const int MOD = 1000000007;
const int INF = 1010000000;
const double EPS = 1e-10;

#define dump(x)  cerr << #x << " = " << (x) << endl; 
#define debug(x) cerr << #x << " = " << (x) << " (L" << __LINE__ << ")" << " " << __FILE__ << endl; 


typedef pair<int, int> P;
struct Point{
	int x, y;
};
int dx[4] = { 0, 1, 0, -1 };
int dy[4] = {1,0,-1,0};



int N;
int X[1001], Y[1001];
ll sum[1001], parity[1001];


int main(){

	cin >> N;
	for (int i = 0; i < N; i++){
		cin >> X[i] >> Y[i];
		sum[i] = X[i] + Y[i];
		parity[i] = abs(sum[i]) % 2;
	}

	int s = 0;
	for (int i = 0; i < N; i++){
		s += parity[i];
	}
	// 全部奇数or全部偶数
	if (s == 0){
		cout << 40 << endl;
		for (int i = 0; i < 40;i++) cout << 1 << " ";
		cout << endl;
		for (int i = 0; i < N; i++){
			if (abs(X[i]) + abs(Y[i]) > 40) return 0;
			if (abs(X[i])>0)
				for (int j = 0; j < abs(X[i]); j++) cout << (X[i] >= 0 ? 'R' : 'L');
			if (abs(Y[i])>0)
				for (int j = 0; j < abs(Y[i]); j++) cout << (Y[i] >= 0 ? 'U' : 'D');
			for (int j = 0; j < 40 - abs(X[i]) - abs(Y[i]); j += 2)
				cout << "UD";
			cout << endl;
		}
	}
	else if (s == N){
		cout << 39 << endl;
		for (int i = 0; i < 39; i++) cout << 1 << " ";
		cout << endl;
		for (int i = 0; i < N; i++){
			if (abs(X[i]) + abs(Y[i]) > 40) return 0;
			if (abs(X[i])>0)
				for (int j = 0; j < abs(X[i]); j++) cout << (X[i] >= 0 ? 'R' : 'L');
			if (abs(Y[i])>0)
				for (int j = 0; j < abs(Y[i]); j++) cout << (Y[i] >= 0 ? 'U' : 'D');
			for (int j = 0; j < 39 - abs(X[i]) - abs(Y[i]); j += 2)
				cout << "UD";
			cout << endl;
		}

	}
	else
		cout << -1 << endl;


	return 0;
}