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

import (
	"bufio"
	"fmt"
	"os"
	"strconv"
	"strings"
)

func getNextLine(scanner *bufio.Reader) string {
	var buffer string
	for {
		line, isPrefix, _ := scanner.ReadLine()
		buffer += string(line)
		if isPrefix == false {
			break
		}
	}
	return buffer
}

func getIntList(scanner *bufio.Reader) []int {
	list := strings.Split(getNextLine(scanner), " ")
	l := len(list)
	result := make([]int, l)
	for i := 0; i < l; i++ {
		result[i], _ = strconv.Atoi(list[i])
	}
	return result
}
func absint(a int) int {
	if a < 0 {
		return -a
	}
	return a
}
func main() {
	fp := os.Stdin
	if len(os.Args) > 1 {
		fp, _ = os.Open(os.Args[1])
	}

	scanner := bufio.NewReader(fp)

	var n int

	fmt.Sscan(getNextLine(scanner), &n)
	xxx := make([]int, n)
	yyy := make([]int, n)

	prev := 0
	current := 0
	max := 0

	for i := 0; i < n; i++ {
		fmt.Sscan(getNextLine(scanner), &xxx[i], &yyy[i])

		abs := absint(xxx[i]) + absint(yyy[i])
		if abs > max {
			max = abs
		}
		current = ((xxx[i]+yyy[i])%2 + 2) % 2
		if i > 0 && prev != current {
			fmt.Println(-1)
			return
		}
		prev = current
	}

	d := 1
	for d < max {
		d <<= 1
	}
	ddd := make([]int, 0)
	if current == 0 {
		ddd = append(ddd, 1)
	}
	for i := 1; i < d; i <<= 1 {
		ddd = append(ddd, i)
	}
	dString := ""
	for i := len(ddd) - 1; i >= 0; i-- {
		dString += fmt.Sprintf("%d", ddd[i])
		if i > 0 {
			dString += " "
		}
	}
	fmt.Println(len(ddd))
	fmt.Println(dString)
	for i := 0; i < n; i++ {
		directions := ""
		for di := len(ddd) - 1; di >= 0; di-- {
			if absint(xxx[i]) > absint(yyy[i]) {
				if xxx[i] > 0 {
					xxx[i] -= ddd[di]
					directions += "R"
					continue
				}
				xxx[i] += ddd[di]
				directions += "L"
				continue
			}
			if yyy[i] > 0 {
				yyy[i] -= ddd[di]
				directions += "U"
				continue
			}
			directions += "D"
			yyy[i] += ddd[di]
		}
		if xxx[i] > 0 {
			directions += "R"
		}
		if xxx[i] < 0 {
			directions += "L"
		}
		if yyy[i] > 0 {
			directions += "U"
		}
		if yyy[i] < 0 {
			directions += "D"
		}
		fmt.Println(directions)
	}
}