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: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
using static System.Math;

public static class P
{
    public static void Main()
    {
        int n = int.Parse(Console.ReadLine());
        var points = Enumerable.Repeat(0, n).Select(_ => Console.ReadLine().Split().Select(int.Parse).ToArray()).Select(x => (x[0], x[1])).ToArray();
        {
            var count = points.Count(x => (x.Item1 + x.Item2) % 2 == 0);
            if (count != 0 && count != n)
            {
                Console.WriteLine(-1);
                return;
            }
        }
        var parity = (points[0].Item1 + points[0].Item2) % 2;
        List<int> arms = new List<int>();
        for (int i = 30; i >= 0; i--)
            arms.Add(1 << i);
        if (parity == 0) arms.Add(1);
        Console.WriteLine(arms.Count);
        Console.WriteLine(string.Join(" ", arms));
        (int, int) getPos(string s)
        {
            int y = 0;
            int x = 0;
            for (int i = 0; i < s.Length; i++)
            {
                switch (s[i])
                {
                    case 'U':
                        y += arms[i];
                        break;
                    case 'D':
                        y -= arms[i];
                        break;
                    case 'R':
                        x += arms[i];
                        break;
                    case 'L':
                        x -= arms[i];
                        break;
                }
            }
            return (y, x);
        }
        foreach (var (x, y) in points)
        {
            long curY = y, curX = x;
            string res = "";
            foreach (var arm in arms)
            {
                var min = long.MaxValue;
                char minOp = '#';
                long minY = 0, minX = 0;
                void Update(long y, long x, char c)
                {
                    var diff = Abs(y) + Abs(x);
                    if (diff < min) { minOp = c; minY = y; minX = x; min = diff; }
                }
                Update(curY - arm, curX, 'U');
                Update(curY + arm, curX, 'D');
                Update(curY, curX - arm, 'R');
                Update(curY, curX + arm, 'L');
                (curY, curX) = (minY, minX);
                res += minOp;
            }
            var pos = getPos(res);
            if (res.Length != arms.Count || pos != (y, x)) throw new Exception();
            Console.WriteLine(res);
        }
    }
}