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: Time Limit Exceeded)

use std::io::stdin;
use std::str::FromStr;

fn main(){
    let n: usize = get_one();
    let xys = get_xys(n);

    let m = if (xys[0].0.abs() + xys[0].1.abs()) % 2 == 0 { 40 } else { 39 };

    let ok = xys.iter().fold(true, |acc, &(x, y)| acc && (x.abs() + y.abs()) % 2 == m % 2);

    if !ok {
        println!("-1");
    } else {
        println!("{}", m);
        for i in 0..m {
            print!("1");
            if i+1 != m {
                print!(" ");
            }
        }
        println!("");

        for i in 0..n {
            for _ in 0..xys[i].0.abs() {
                print!("{}", if xys[i].0 < 0 { 'L' } else { 'R' });
            }
            for _ in 0..xys[i].1.abs() {
                print!("{}", if xys[i].1 < 0 { 'D' } else { 'U' });
            }
            for _ in 0..(m-xys[i].0.abs()-xys[i].1.abs()) / 2 {
                print!("LR");
            }
            println!("");
        }
    }
}

fn get_xys(n: usize) -> Vec<(i64, i64)> {
    let mut xys = Vec::new();
    for _ in 0..n {
        xys.push(get_pair());
    }
    xys
}

#[allow(dead_code)]
fn get_line() -> String {
    let mut s = String::new();
    match stdin().read_line(&mut s){
        Ok(_) => {s.trim().to_string()}
        Err(_) => String::new()
    }
}

#[allow(dead_code)]
fn get_vec<T: std::str::FromStr>() -> Vec<T> {
    let line = get_line();
    line.split_whitespace().filter_map(|x| x.parse().ok()).collect()
}

#[allow(dead_code)]
fn get_one<T: FromStr + Copy>() -> T {
    let v = get_vec();
    v[0]
}

#[allow(dead_code)]
fn get_pair<T: FromStr + Copy>() -> (T, T) {
    let v = get_vec();
    (v[0], v[1])
}

#[allow(dead_code)]
fn get_triple<T: FromStr + Copy>() -> (T, T, T) {
    let v = get_vec();
    (v[0], v[1], v[2])
}

#[allow(dead_code)]
fn get_chars() -> Vec<char> {
    get_line().chars().collect()
}