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)

DBG = !true
n = gets.to_i
a = []
even1 = false
odd1 = false
for i in 0...n
  x,y = gets.split.map{|z| z.to_i}
  a << [x, y]
  if (x+y)%2 == 0
    even1 = true
  else
    odd1 = true
  end
end

if odd1 && even1
  print -1
  exit
end

# if odd, add one to y
addstr = ''
if even1
  a.map!{|x| [x[0], x[1]+1]}
  addstr = 'D'
end

ds = []
d = 1
0.upto(35) {
  ds << d
  d *= 2
}
szd = 36
if even1
  ds << 1
  szd += 1
end

print szd
puts ""
for i in 0...szd
  print ds[i], " "
end
puts ""

for k in 0...n
  x,y = a[k]
  s = "012345678901234567890123456789012345" + addstr
  cx = cy = 0
  35.downto(0) { |i|
    dx = x-cx
    dy = y-cy
    if dx+dy > 0 && dx-dy > 0
      s[i] = 'R'
      cx += ds[i]
    elsif dx+dy < 0 && dx-dy > 0
      s[i] = 'D'
      cy -= ds[i]
    elsif dx+dy > 0 && dx-dy < 0
      s[i] = 'U'
      cy += ds[i]
    else #if dx+dy < 0 && dx-dy < 0
      s[i] = 'L'
      cx -= ds[i]
    end
  }
  puts s

  if DBG
    cx = cy = 0
    for i in 0...szd
      ch = s[i]
      if ch == 'R'
        cx += ds[i]
      elsif ch == 'L'
        cx -= ds[i]
      elsif ch == 'U'
        cy += ds[i]
      elsif ch == 'D'
        cy -= ds[i]
      end
    end
    realy = y - (even1 ? 1 : 0)
    if cx==x && cy==realy
      puts "ok"
    else
      printf("ng: x %d y %d cx %d cy %d\n", x, y, cx, cy)
    end
  end
end