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 JavaScript (node.js v5.12) (judge verdict: Runtime Error)

var toInt = x=>parseInt(x,10);
var toIntArr = arr=>arr.map(x=>toInt(x));
var abs = x=>x>0?x:-x;
function min(){var rest=arguments,v=rest[0];for(var i=1;i<rest.length;i++)if(rest[i]<v)v=rest[i];return v}
function max(){var rest=arguments,v=rest[0];for(var i=1;i<rest.length;i++)if(rest[i]>v)v=rest[i];return v}
function gcd(){var rest=arguments,v=rest[0],temp=(a,b)=>b?temp(b,a%b):a;for(var i=1;i<rest.length;i++)v=temp(v,rest[i]);return v}
Main(require("fs").readFileSync("/dev/stdin", "utf8"));
 
function Main(input){
	var input = input.split('\n');
	var n = toInt(input[0]);
	var xys = [];
	for(var i = 0; i < n; i++){
		xys[i] = toIntArr(input[i+1].split(' '));
		if((xys[i][0]+xys[i][1]+xys[0][0]+xys[0][1])%2 !== 0){
			console.log(-1);
			return 0;
		}
	}
	var po = (xys[0][0]+xys[0][1])%2;
	if(po===1){
		xys = xys.map(e=>e[0]%2===0?e:e.reverse());
	}
	/*
	even
	m = 10
	di = 1*10
	2n -> 4:+ (10-4)/2:+,-	
	*/
	var m = 20;
	if(xys[0][0]%2===1) m++;
	if(xys[0][1]%2===1) m++;
	console.log(m);
	var ds = [];
	ds.length = m;
	ds.fill(1);
	ds = ds.join(' ');
	console.log(ds);

	for(var i = 0; i < n; i++){
		var str = '';
		var xy = xys[i];
		if(xy[0]%2===1){
			str += 'R';
			xy[0]++;
		}
		if(xy[1]%2===1){
			str += 'R';
			xy[1]++;
		}
	str += ''.padEnd((10+xy[0])/2,'R');
	str += ''.padEnd((10-xy[0])/2,'L');
	str += ''.padEnd((10+xy[1])/2,'U');
	str += ''.padEnd((10-xy[1])/2,'D');
	console.log(str);
	}

	return 0;
}