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"
)

const size = 1000000
var reader = bufio.NewReaderSize(os.Stdin,size)
func NextLine() string {
  var line,buffer []byte
  var isPrefix bool = true
  var err error
  for ;isPrefix; {
    line,isPrefix,err = reader.ReadLine()
    if err != nil { panic(err) }
    buffer = append(buffer,line...)
  }
  return string(buffer)
}
func NextInt() int {
  var x int
  x,_ = strconv.Atoi(NextLine())
  return x
}
func Next2Ints() (int,int) {
  var x,y int
  var s []string = strings.Split(NextLine()," ")
  x,_ = strconv.Atoi(s[0])
  y,_ = strconv.Atoi(s[1])
  return x,y
}

func main() {
  var n int = NextInt()
  var x []int = make([]int,n)
  var y []int = make([]int,n)
  var u []int64 = make([]int64,n)
  var v []int64 = make([]int64,n)
  for i:=0;i<n;i++ {
    x[i],y[i] = Next2Ints()
    u[i] = int64(x[i]) + int64(y[i])
    v[i] = int64(x[i]) - int64(y[i])
  }
  var p int64 = abs(u[0])%2
  for i:=1;i<n;i++ {
    if abs(u[i])%2 != p {
      fmt.Println(-1)
      os.Exit(0)
    }
  }
  var m int = 31
  var d []int64 = make([]int64,m)
  d[0] = 1
  for i:=1;i<m;i++ {
    d[i] = 2*d[i-1]
  }
  if p == 0 {
    d = append(d,1)
    for i:=0;i<n;i++ {
      u[i]--
      v[i]--
    }
  }
  fmt.Println(m+int(1-p))
  fmt.Print(d[0])
  for i:=1;i<len(d);i++ {
    fmt.Print(" ")
    fmt.Print(d[i])
  }
  fmt.Println()
  var max int64 = 2*d[m-1]-1
  for i:=0;i<n;i++ {
    var us string = strconv.FormatInt((u[i]+max)/2,2)
    var vs string = strconv.FormatInt((v[i]+max)/2,2)
    var t string = ""
    for {
      if len(us) == m { break }
      us = "0" + us
    }
    for {
      if len(vs) == m { break }
      vs = "0" + vs
    }
    us = reverse(us)
    vs = reverse(vs)
    for i:=0;i<m;i++ {
      if us[i:i+1] == "0" && vs[i:i+1] == "0" {
        t += "L"
      } else if us[i:i+1] == "0" && vs[i:i+1] == "1" {
        t += "D"
      } else if us[i:i+1] == "1" && vs[i:i+1] == "0" {
        t += "U"
      } else {
        t += "R"
      }
    }
    if p == 0 { t += "R" }
    fmt.Println(t)
  }
}
func abs(n int64) int64 {
  if n < 0 { return -n }
  return n
}
func reverse(s string) string {
  var r []rune = []rune(s)
  var n int = len(s)
  for i:=0;i<n-i;i++ {
    r[i],r[n-i-1] = r[n-i-1],r[i]
  }
  return string(r)
}