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 Java (OpenJDK 11.0.6) (judge verdict: Accepted)

import java.util.Arrays;
import java.util.Scanner;
import java.util.stream.Collectors;
import java.util.stream.IntStream;

public class Main {
    static final char[] DIRECTIONS = { 'L', 'R', 'D', 'U' };
    static final int[] X_OFFSETS = { -1, 1, 0, 0 };
    static final int[] Y_OFFSETS = { 0, 0, -1, 1 };
    static final int REGULATION_MOVE = 0;

    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);

        int N = sc.nextInt();
        int[] X = new int[N];
        int[] Y = new int[N];
        for (int i = 0; i < N; ++i) {
            X[i] = sc.nextInt();
            Y[i] = sc.nextInt();
        }

        System.out.println(solve(X, Y));

        sc.close();
    }

    static String solve(int[] X, int[] Y) {
        if (IntStream.range(0, X.length).map(i -> Math.abs((X[i] + Y[i]) % 2)).distinct().count() != 1) {
            return "-1";
        }

        boolean regulationNeeded = (X[0] + Y[0]) % 2 == 0;

        if (regulationNeeded) {
            for (int i = 0; i < X.length; ++i) {
                X[i] -= X_OFFSETS[REGULATION_MOVE];
                Y[i] -= Y_OFFSETS[REGULATION_MOVE];
            }
        }

        int[] d = IntStream.range(0, 31).map(i -> 1 << i).toArray();

        String[] w = new String[X.length];
        for (int i = 0; i < w.length; ++i) {
            int x = X[i];
            int y = Y[i];
            char[] path = new char[d.length];
            for (int j = path.length - 1; j >= 0; --j) {
                for (int k = 0;; ++k) {
                    int nextX = x - X_OFFSETS[k] * d[j];
                    int nextY = y - Y_OFFSETS[k] * d[j];

                    if ((long) Math.abs(nextX) + Math.abs(nextY) <= (1 << j) - 1) {
                        x = nextX;
                        y = nextY;
                        path[j] = DIRECTIONS[k];

                        break;
                    }
                }
            }

            w[i] = new String(path);
        }

        if (regulationNeeded) {
            d = Arrays.copyOf(d, d.length + 1);
            d[d.length - 1] = 1;

            for (int i = 0; i < w.length; ++i) {
                w[i] += DIRECTIONS[REGULATION_MOVE];
            }
        }

        return String.format("%d\n%s\n%s", d.length,
                Arrays.stream(d).mapToObj(String::valueOf).collect(Collectors.joining(" ")), String.join("\n", w));
    }
}