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::{self, BufRead};
use std::collections::BTreeSet;
use Dir::*;

#[derive(Copy, Clone)]
enum Dir {
  Up, Down, Left, Right
}

impl Dir {
  fn to_char(self) -> u8 {
    match self {
      Up => 'U' as u8,
      Down => 'D' as u8,
      Left => 'L' as u8,
      Right => 'R' as u8,
    }
  }
}

fn main() {
  let stdin = io::stdin();
  let mut lines = stdin.lock().lines();
  let n: usize = lines.next().unwrap().unwrap().parse().unwrap();
  let mut ps: Vec<(i32, i32)> = Vec::new();
  for _ in 0..n {
    let xs: Vec<i32> = lines.next().unwrap().unwrap().split(' ').map(|x| x.parse().unwrap()).collect();
    ps.push((xs[0], xs[1]));
  }
  let even = ps.iter().filter(|p| (p.0.abs() + p.1.abs()) % 2 == 0).count();
  if even != 0 && even != ps.len() {
    println!("-1");
    return;
  }
  let len = ps.iter().map(|p| p.0.abs() + p.1.abs()).max().unwrap();
  let mut joints: BTreeSet<i32> = BTreeSet::new();
  for &(px, py) in &ps {
    if px != 0 && py != 0 {
      joints.insert(px.abs() as i32);
    }
    let dist = px.abs() + py.abs();
    if dist != len {
      joints.insert((dist + len) / 2);
    }
  }
  let joints: Vec<i32> = joints.into_iter().collect();
  let mut temp = 0;
  for i in 0..joints.len() {
    print!("{} ", joints[i] - temp);
    temp = joints[i];
  }
  println!("{}", if joints.is_empty() { len } else { len - joints[joints.len()-1] });
  for (px, py) in ps {
    let mut d = if px == 0 {
      if py >= 0 { Up } else { Down }
    } else if px >= 0 {
      Right
    } else {
      Left
    };
    let mut ds: Vec<u8> = vec![d.to_char()];
    let p1 = px.abs();
    let p2 = py.abs();
    let p3 = (len + p1 + p2) / 2;
    for i in 0..joints.len() {
      if p1 == joints[i] {
        d = if py >= 0 { Up } else { Down };
      } else if p3 == joints[i] {
        d = match d {
          Up => Down,
          Down => Up,
          Left => Right,
          Right => Left,
        };
      }
      ds.push(d.to_char());
    }
    println!("{}", String::from_utf8(ds).unwrap());
  }
}