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: Accepted)

#![allow(non_snake_case, unused_macros, unused_imports)]

// Input
macro_rules! input { (source = $s:expr, $($r:tt)*) => { let mut iter = $s.split_whitespace(); let mut next = || { iter.next().unwrap() }; input_inner!{next, $($r)*} }; ($($r:tt)*) => { let stdin = std::io::stdin(); let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock())); let mut next = move || -> String{ bytes.by_ref().map(|r|r.unwrap() as char).skip_while(|c|c.is_whitespace()).take_while(|c|!c.is_whitespace()).collect() }; input_inner!{next, $($r)*} }; }
macro_rules! input_inner { ($next:expr) => {}; ($next:expr, ) => {}; ($next:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($next, $t); input_inner!{$next $($r)*} };}
macro_rules! read_value { ($next:expr, ( $($t:tt),* )) => { ( $(read_value!($next, $t)),* ) }; ($next:expr, [ $t:tt ; $len:expr ]) => { (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>() }; ($next:expr, chars) => { read_value!($next, String).chars().collect::<Vec<char>>() }; ($next:expr, bytes) => { read_value!($next, String).into_bytes() }; ($next:expr, usize1) => { read_value!($next, usize) - 1 }; ($next:expr, $t:ty) => { $next().parse::<$t>().expect("Parse error") }; }

// Module
use std::cmp::{min,max};
use std::collections::{HashSet,BTreeMap};

// Main
fn main(){
    input!{
        n: usize,
        xy: [(i64,i64); n],
    }
    let mut xy = xy;
    // パリティのチェック
    let mut even = 0;
    let mut odd = 0;
    for i in 0..n{
        if (xy[i].0 + xy[i].1)%2==0{
            even += 1;
        }else{
            odd += 1;
        }
    }
    // パリティが0,1混在の場合は実現不可能
    if even>0 && odd>0 {
        println!("{}", -1);
        return;
    }
    // パリティが0の場合 → X座標を-1することでパリティが1の場合に帰着、後で調整必要
    if even>0 {
        for i in 0..n{
            xy[i].0 -= 1;
        }
    }
    // アームを準備
    let mut arms = vec![];
    for i in 0..35 {
        arms.push(2_i64.pow(i));
    }
    arms.sort_by(|a,b| b.cmp(a));
    // 順番に処理
    let mut Ans: Vec<Vec<char>> = vec![];
    for i in 0..n {
        let mut ans: Vec<char> = vec![];
        let (mut x, mut y) = (xy[i].0, xy[i].1);
        let (mut x0, mut y0) = (0,0);
        for arm in &arms {
            let tx = x - x0;
            let ty = y - y0;
            if &ty < &tx && &ty > &-tx {
                ans.push('R');
                x0 += *arm;
            }else if &ty<&tx && &ty<&-tx {
                ans.push('D');
                y0 -= *arm;
            }else if &ty>&tx && &ty<&-tx{
                ans.push('L');
                x0 -= *arm;
            }else{
                ans.push('U');
                y0 += *arm;
            }
        }
        Ans.push(ans);
    }
    // パリティが0の場合、長さ1のアームを追加して、操作の最後にRを追加
    if even>0 {
        arms.push(1);
        println!("{}", arms.len());
        for arm in arms {
            print!("{} ", arm);
        }
        println!();
        for ans in Ans {
            for c in ans {
                print!("{}", c);
            }
            println!("{}", 'R');
        }
    }else{
        println!("{}", arms.len());
        for arm in arms {
            print!("{} ", arm);
        }
        println!();
        for ans in Ans {
            for c in ans {
                print!("{}", c);
            }
            println!();
        }
    }
}