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 Rust (1.15.1) (judge verdict: Wrong Answer)

use std::io;
use std::io::Read;

fn main() {
    // inputs
    let mut stdin = io::stdin();
    let mut lines_str = String::new();
    let _ = stdin.read_to_string(&mut lines_str);
    let mut lines = lines_str.lines();
    let n :usize = lines.next().expect("").trim().parse().unwrap();
    let mut xy :Vec<(isize,isize)> = Vec::new();
    for _ in 0..n {
        let mut s = lines.next().expect("").split_whitespace();
        let x = s.next().expect("").trim().parse().unwrap();
        let y = s.next().expect("").trim().parse().unwrap();
        xy.push((x,y));
    }
    let xy = xy;

    let parity = (xy[0].0 + xy[0].1).abs() % 2;
    //parity check and size check
    for &(x,y) in &xy {
        if (x+y).abs() % 2 != parity {
            println!("-1");
            return;
        }
        if x.abs() > 10 {
            return;
        }
        if y.abs() > 10 {
            return;
        }
    }

    let m;
    if parity == 0 {
        m = 20;
    } else {
        m = 19;
    }

    // outputs
    println!("{}",m);
    if parity == 0 {
        println!("1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1");
    } else {
        println!("1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1");
    }
    for &(x,y) in &xy {
        let mut xx = x;
        let mut yy = y;
        let mut mm = m;
        //x axis
        while xx != 0 {
            if xx > 0 {
                print!("R");
                xx -= 1;
                mm -= 1;
            } else {
                print!("L");
                xx += 1;
                mm -= 1;
            }
        }
        //y axis
        while yy != 0 {
            if yy > 0 {
                print!("U");
                yy -= 1;
                mm -= 1;
            } else {
                print!("D");
                yy += 1;
                mm -= 1;
            }
        }
        while mm != 0 {
            print!("UD");
            mm -= 2;
        }
        println!("");
    }
}