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 Go (1.6) (judge verdict: Accepted)

// Package main provides
//
// File:  d.go
// Author: ymiyamoto
//
// Created on Wed Dec 19 22:42:09 2018
//
package main

import "fmt"

type point struct {
	x, y int
}

func abs(a int) int {
	if a < 0 {
		return -a
	}
	return a
}

func checkPossible(points []point) bool {
	oe := abs(points[0].x+points[0].y) % 2
	for _, p := range points {
		if oe != abs(p.x+p.y)%2 {
			return false
		}
	}
	return true
}

func pow2(n int) int {
	if n == 0 {
		return 1
	}
	return 2 * pow2(n-1)
}

func main() {
	var N int
	fmt.Scan(&N)
	points := make([]point, N)
	for i := range points {
		fmt.Scan(&points[i].x, &points[i].y)
	}

	if !checkPossible(points) {
		fmt.Println(-1)
		return
	}
	arms := []int{}
	if abs(points[0].x+points[0].y)%2 == 0 {
		arms = append(arms, 1)
	}

	for i := 0; i < 39; i++ {
		arms = append(arms, pow2(i))
	}

	fmt.Println(len(arms))
	for i, arm := range arms {
		if i != 0 {
			fmt.Print(" ")
		}
		fmt.Print(arm)
	}
	fmt.Println()

	for _, p := range points {
		u := p.x + p.y
		v := p.x - p.y
		modes := ""
		for i := len(arms) - 1; i >= 0; i-- {
			l := arms[i]
			if u >= 0 && v >= 0 {
				u -= l
				v -= l
				modes += "R"
			} else if u >= 0 && v <= 0 {
				u -= l
				v += l
				modes += "U"
			} else if u <= 0 && v >= 0 {
				u += l
				v -= l
				modes += "D"
			} else {
				u += l
				v += l
				modes += "L"
			}
		}
		for i := len(modes) - 1; i >= 0; i-- {
			fmt.Print(string(modes[i]))
		}
		fmt.Println()
	}
}