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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.*;

public class Main {
    int dx[] = { 0, 1, 0, -1 };
    int dy[] = { 1, 0, -1, 0 };
    char dr[] = "DLUR".toCharArray();

    private class Point {
        long x, y;

        public Point(int x, int y) {
            this.x = x;
            this.y = y;
        }

        public boolean isOdd() {
            return (x + y) % 2 == 0;
        }

        public long dist() {
            return Math.abs(x) + Math.abs(y);
        }
    }

    public void main(Scanner sc) {
        int n = sc.nextInt();
        Point p[] = new Point[n];
        BitSet bs = new BitSet();
        long max = 0;

        for (int i = 0; i < n; i++) {
            p[i] = new Point(sc.nextInt(), sc.nextInt());
            if (p[i].isOdd()) {
                bs.set(i);
            }
            max = Math.max(max, p[i].dist());
        }
        sc.close();

        if (bs.cardinality() > 0 && bs.cardinality() < n) {
            System.out.println(-1);
            return;
        }

        boolean isOdd = bs.cardinality() == n;

        int m = 0;
        for (; max > 1 << m; m++) {
        }
        long l[] = new long[m + 1];
        for (int i = 0; i < m; i++) {
            l[i + 1] = 1L << i;
        }

        if (isOdd) {
            for (int i = 0; i < n; i++) {
                p[i].x--;
            }
            System.out.println(m + 1);
        } else {
            System.out.println(m);
        }

        StringBuilder sb = new StringBuilder(isOdd ? "1 " : "");
        for (int i = 1; i <= m; i++) {
            sb.append(l[i] + " ");
        }
        System.out.println(sb.toString().trim());

        for (int i = 0; i < n; i++) {
            char move[] = new char[m];
            long nx = p[i].x;
            long ny = p[i].y;

            for (int j = m; j > 0; j--) {
                for (int k = 0; k < 4; k++) {
                    if (check(nx + l[j] * dx[k], ny + l[j] * dy[k], l[j - 1])) {
                        nx += l[j] * dx[k];
                        ny += l[j] * dy[k];
                        move[j - 1] = dr[k];
                        break;
                    }
                }
            }

            if (isOdd) {
                System.out.print("R");
            }
            System.out.println(new String(move));
        }
    }

    private boolean check(long x, long y, long range) {
        return Math.abs(x) + Math.abs(y) <= range;
    }

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