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: Wrong Answer)

package main

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

type Point struct {
	x int
	y int
}

func main() {
	sc := bufio.NewScanner(os.Stdin)
	sc.Split(bufio.ScanWords)

	nPoints := scanInt(sc)
	first := Point{
		x: scanInt(sc),
		y: scanInt(sc),
	}

	maxD := abs(first.x) + abs(first.y)
	parity := maxD % 2

	ps := make([]Point, nPoints)
	ps[0] = first
	for i := 1; i < nPoints; i++ {
		p := Point{
			x: scanInt(sc),
			y: scanInt(sc),
		}
		ps[i] = p

		d := abs(p.x) + abs(p.y)
		if parity != d%2 {
			fmt.Println(-1)
			return
		}

		if d > maxD {
			maxD = d
		}
	}

	nArms := uint(math.Ceil(math.Log2(float64(maxD))))
	if maxD == 0 {
		nArms = 2 // 腕は 1 本以上必要
	}

	out := bufio.NewWriter(os.Stdout)

	fmt.Fprintln(out, nArms)
	for i := nArms; i >= 1; i-- {
		fmt.Fprintf(out, "%d ", 1<<(i-1))
	}
	if parity == 0 {
		fmt.Fprintf(out, "%d ", 1)
	}
	fmt.Fprintln(out)

	adjust := parity - 1
	for _, p := range ps {
		x := p.x
		y := p.y + adjust
		for exp := nArms; exp >= 1; exp-- {
			a := int(1 << (exp - 1))
			if abs(x) > abs(y) {
				if x > 0 {
					fmt.Fprint(out, "R")
					x -= a
				} else if x < 0 {
					fmt.Fprint(out, "L")
					x += a
				}
			} else {
				if y > 0 {
					fmt.Fprint(out, "U")
					y -= a
				} else if y < 0 {
					fmt.Fprint(out, "D")
					y += a
				}
			}
		}
		if adjust == -1 {
			fmt.Fprint(out, "U")
		}
		fmt.Fprintln(out)
	}
	out.Flush()
}

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

func scanInt(sc *bufio.Scanner) int {
	sc.Scan()
	v, _ := strconv.Atoi(sc.Text())
	return v
}