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# (.NET Core 3.1.201) (judge verdict: Wrong Answer)

using System;
using System.Linq;

class D
{
	static int[] Read() => Console.ReadLine().Split().Select(int.Parse).ToArray();
	static void Main()
	{
		var n = int.Parse(Console.ReadLine());
		var ps = new int[n].Select(_ => Read()).Select(v => (x: v[0], y: v[1])).ToArray();
		if (ps.Select(p => Math.Abs(p.x + p.y) % 2).Distinct().Count() == 2) { Console.WriteLine(-1); return; }

		var ds = new int[41];
		var uy = new bool[n, 41]; // for y-direction

		var qs = ps.Select((p, i) =>
		{
			var (x, y) = p;
			(x, y) = (Math.Abs(x), Math.Abs(y));
			uy[i, 0] = x < y;
			return x < y ? (x: y, y: x) : (x, y);
		}).ToArray();

		// Repeat to halve `max(x + y)`
		ds[0] = qs.Max(p => p.x);

		for (int k = 0; k < 40; k++)
		{
			qs = qs.Select((p, i) =>
			{
				var (x, y) = p;
				x = Math.Abs(x - ds[k]);
				uy[i, k + 1] = (x < y) ^ uy[i, k];
				return x < y ? (y, x) : (x, y);
			}).ToArray();

			var min = qs.Min(p => p.x);
			var max = qs.Max(p => p.x);
			if (max == 0) break;
			ds[k + 1] = (min + max + 1) / 2;
		}

		var m = ds.TakeWhile(d => d > 0).Count();
		Console.WriteLine(m);
		Console.WriteLine(string.Join(" ", ds[..m]));

		var rs = ps.Select((p, i) =>
		{
			var (x, y) = p;
			var c = new char[m];

			for (int k = 0; k < m; k++)
			{
				if (uy[i, k])
				{
					if (y < 0)
					{
						y += ds[k];
						c[k] = 'D';
					}
					else
					{
						y -= ds[k];
						c[k] = 'U';
					}
				}
				else
				{
					if (x < 0)
					{
						x += ds[k];
						c[k] = 'L';
					}
					else
					{
						x -= ds[k];
						c[k] = 'R';
					}
				}
			}
			return new string(c);
		});
		Console.WriteLine(string.Join("\n", rs));
	}
}