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# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
public struct Target
{
    public int x, y, sx, sy, d1, d2, symFlag;
}
class hogehoge
{
	static void Main(string[] args)
	{
		int n = int.Parse(Console.ReadLine());
		Target[] t = new Target[n];
		for (int i = 0; i < n; i++)
		{
			string[] s = Console.ReadLine().Split(' ');
			t[i].x = int.Parse(s[0]);
			t[i].y = int.Parse(s[1]);
			t[i].sx = Math.Abs(t[i].x);
			if (t[i].sx != t[i].x) t[i].symFlag += 1;
			t[i].sy = Math.Abs(t[i].y);
			if (t[i].sy != t[i].y) t[i].symFlag += 2;
			if (t[i].sy > t[i].sx)
			{
				int buf = t[i].sx;
				t[i].sx = t[i].sy;
				t[i].sy = buf;
				t[i].symFlag += 4;
			}
			t[i].d1 = t[i].sx + t[i].sy;
			t[i].d2 = t[i].sx - t[i].sy;
		}
		bool isEven, isPossible = true;
		if (t[0].d1 % 2 == 0) isEven = true;
		else isEven = false;
		for (int i = 1; i < n; i++)
		{
			if (isEven && t[i].d1 % 2 == 1)
			{
				isPossible = false;
				break;
			}
			if (!isEven && t[i].d1 % 2 == 0)
			{
				isPossible = false;
				break;
			}
		}
		if (!isPossible)
		{
			Console.WriteLine(-1);
		}
		else
		{
			if (isEven)
			{
				Console.WriteLine(32);
				string m = "";
				for (int i = 1073741824; i > 1; i /= 2)
				{
					m += i + " ";
				}
				m += "1 1";
				Console.WriteLine(m);
				for (int i = 0; i < n; i++)
				{
					string w = "";
					int p1 = 0, p2 = 0;
					for (int d = 1073741824; d > 1; d /= 2)
					{
						if (t[i].d1 <= p1 && t[i].d2 <= p2)
						{
							w += "l";
							p1 -= d;
							p2 -= d;
						}
						else if (t[i].d1 <= p1 && t[i].d2 > p2)
						{
							w += "d";
							p1 -= d;
							p2 += d;
						}
						else if (t[i].d1 > p1 && t[i].d2 > p2)
						{
							w += "r";
							p1 += d;
							p2 += d;
						}
						else if (t[i].d1 > p1 && t[i].d2 <= p2)
						{
							w += "u";
							p1 += d;
							p2 -= d;
						}
					}
					if (t[i].symFlag == 0)
					{
						w.Replace("l", "L");
						w.Replace("r", "R");
						w.Replace("d", "D");
						w.Replace("u", "U");
					}
					else if (t[i].symFlag == 1)
					{
						w.Replace("l", "R");
						w.Replace("r", "L");
						w.Replace("d", "D");
						w.Replace("u", "U");
					}
					else if (t[i].symFlag == 2)
					{
						w.Replace("l", "L");
						w.Replace("r", "R");
						w.Replace("d", "U");
						w.Replace("u", "D");
					}
					else if (t[i].symFlag == 3)
					{
						w.Replace("l", "R");
						w.Replace("r", "L");
						w.Replace("d", "U");
						w.Replace("u", "D");
					}
					else if (t[i].symFlag == 4)
					{
						w.Replace("l", "D");
						w.Replace("r", "U");
						w.Replace("d", "L");
						w.Replace("u", "R");
					}
					else if (t[i].symFlag == 5)
					{
						w.Replace("l", "U");
						w.Replace("r", "D");
						w.Replace("d", "L");
						w.Replace("u", "R");
					}
					else if (t[i].symFlag == 6)
					{
						w.Replace("l", "D");
						w.Replace("r", "U");
						w.Replace("d", "R");
						w.Replace("u", "L");
					}
					else if (t[i].symFlag == 7)
					{
						w.Replace("l", "U");
						w.Replace("r", "D");
						w.Replace("d", "R");
						w.Replace("u", "L");
					}
					Console.WriteLine(w);
				}
			}
			else
			{
				Console.WriteLine(32);
				string m = "1073741823 ";
				for (int i = 1073741824 / 2; i > 1; i /= 2)
				{
					m += i + " ";
				}
				m += "1 1";
				Console.WriteLine(m);
				for (int i = 0; i < n; i++)
				{
					string w = "r";
					int p1 = 1073741823, p2 = 1073741823;
					for (int d = 1073741824 / 2; d > 1; d /= 2)
					{
						if (t[i].d1 <= p1 && t[i].d2 <= p2)
						{
							w += "l";
							p1 -= d;
							p2 -= d;
						}
						else if (t[i].d1 <= p1 && t[i].d2 > p2)
						{
							w += "d";
							p1 -= d;
							p2 += d;
						}
						else if (t[i].d1 > p1 && t[i].d2 > p2)
						{
							w += "r";
							p1 += d;
							p2 += d;
						}
						else if (t[i].d1 > p1 && t[i].d2 <= p2)
						{
							w += "u";
							p1 += d;
							p2 -= d;
						}
					}
					if (t[i].symFlag == 0)
					{
						w.Replace("l", "L");
						w.Replace("r", "R");
						w.Replace("d", "D");
						w.Replace("u", "U");
					}
					else if (t[i].symFlag == 1)
					{
						w.Replace("l", "R");
						w.Replace("r", "L");
						w.Replace("d", "D");
						w.Replace("u", "U");
					}
					else if (t[i].symFlag == 2)
					{
						w.Replace("l", "L");
						w.Replace("r", "R");
						w.Replace("d", "U");
						w.Replace("u", "D");
					}
					else if (t[i].symFlag == 3)
					{
						w.Replace("l", "R");
						w.Replace("r", "L");
						w.Replace("d", "U");
						w.Replace("u", "D");
					}
					else if (t[i].symFlag == 4)
					{
						w.Replace("l", "D");
						w.Replace("r", "U");
						w.Replace("d", "L");
						w.Replace("u", "R");
					}
					else if (t[i].symFlag == 5)
					{
						w.Replace("l", "U");
						w.Replace("r", "D");
						w.Replace("d", "L");
						w.Replace("u", "R");
					}
					else if (t[i].symFlag == 6)
					{
						w.Replace("l", "D");
						w.Replace("r", "U");
						w.Replace("d", "R");
						w.Replace("u", "L");
					}
					else if (t[i].symFlag == 7)
					{
						w.Replace("l", "U");
						w.Replace("r", "D");
						w.Replace("d", "R");
						w.Replace("u", "L");
					}
					Console.WriteLine(w);
				}
			}
		}
	}
}