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 Ruby (2.3.3) (judge verdict: Accepted)

class Point
  attr_reader :x, :y

  def initialize(x, y)
    @x = x
    @y = y
  end
  def to_s
    "(#{x},#{y})"
  end

  def dist_from_origin
    x.abs + y.abs
  end

  # 腕を減算し、その腕の方向を返す
  def sub_len(len)
    x_abs = @x.abs
    y_abs = @y.abs
    if y_abs <= x_abs
      if 0 <= @x
        @x -= len
        "R"
      else
        @x += len
        "L"
      end
    else
      if 0 <= @y
        @y -= len
        "U"
      else
        @y += len
        "D"
      end
    end
  end
end

def calc_arms(dist)
  arms = []
  # 偶数は面倒なので最初に1引いておく
  arms << 1 if dist.even?
  dist -= 1

  # armは [1, 2, 4, ..., longest_arm] を取り揃える。
  #   dist = longest_arm * 2 - 1
  # までは届く。
  # 最初に追加した1を加えると,
  #   dist = longest_arm * 2
  # まで届く。
  longest_arm = 1
  arms << 1
  while true
    next_arm = longest_arm * 2
    break if dist < next_arm

    longest_arm = next_arm
    arms << longest_arm
  end
  arms
end

def main
  n = gets.chomp.to_i

  is_odd = nil  # 偶数奇数チェック ※すべての距離が偶数 or 奇数 で揃っていないとNG
  max_dist = 0  # 原点からの最長距離
  points = Array.new(n)
  n.times do |i|
    xy = Point.new *gets.split.map(&:to_i)
    dist = xy.dist_from_origin
    max_dist = dist if max_dist < dist
    if is_odd.nil?
      is_odd = dist.odd?
    elsif is_odd != dist.odd?
      puts "-1"
      return
    end

    points[i] = xy
  end
  # puts points

  # armセットを作り上げる
  arms = calc_arms(max_dist).reverse
  puts arms.length
  puts arms.join(" ")

  # 問題を解いていく
  points.each do |p|
    arm_dirs = []

    arms.each do |arm|
      arm_dirs << p.sub_len(arm)
    end

    puts arm_dirs.join
  end
end

if __FILE__ == $0
  main
end