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: Wrong Answer)

n = gets.to_i

list = []
n.times{
	list << gets.strip.split(/ /).collect{|item| item.to_i}
}

def check(list)
	flag = nil
	list.each{|x, y|
		cur_flag = ((x + y) % 2 == 0 ? :even : :odd)
		if flag == nil
			flag = cur_flag
		elsif flag != cur_flag
			return false
		end
	}
	return true
end

def calc_arm_max(max)
	arm_count = 1
	arm_max = 1
	range_max = 1
	while range_max < max
		arm_max *= 2
		range_max += arm_max
		arm_count += 1
	end
	[arm_count, arm_max]
end

def check_arms(arms, w)
	x = 0
	y = 0
	arms.each_with_index{|a, i|
		case w[i]
		when 'L'
			x -= a
		when 'R'
			x += a
		when 'D'
			y -= a
		when 'U'
			y += a
		end
	}
	[x, y]
end

if check(list)
	flag = ((list[0][0] + list[0][1]) % 2 == 0 ? :even : :odd)
	x_max = list.collect{|x, y| x.abs}.max
	y_max = list.collect{|x, y| y.abs}.max
	
	x_arm_count, x_arm_max = calc_arm_max(x_max)
	y_arm_count, y_arm_max = calc_arm_max(y_max)
	
	arms = []
	
	if flag == :even
		arms << 1
		arms << 1
	else
		arms << 1
	end
	
	arm_len = x_arm_max
	x_arm_count.times{
		arms << arm_len
		arm_len /= 2
	}

	arm_len = y_arm_max
	y_arm_count.times{
		arms << arm_len
		arm_len /= 2
	}
	
	puts arms.length
	
	puts arms.join(' ')
	
	list.each{|x, y|
		w = ''
		
		org_x = x
		org_y = y
		
		if flag == :even
			if x % 2 == 0
				if x > 0
					w << 'R'
					x -= 1
				else
					w << 'L'
					x += 1
				end
				if y > 0
					w << 'U'
					y -= 1
				else
					w << 'D'
					y += 1
				end
			else
				w << 'LR'
			end
		else
			if x % 2 == 0
				if x > 0
					w << 'R'
					x -= 1
				else
					w << 'L'
					x += 1
				end
			else
				if y > 0
					w << 'U'
					y -= 1
				else
					w << 'D'
					y += 1
				end
			end
		end
		
		arm_len = x_arm_max
		x_arm_count.times{
			if x > 0
				x -= arm_len
				w << 'R'
			else
				x += arm_len
				w << 'L'
			end
			arm_len /= 2
		}
		
		arm_len = y_arm_max
		y_arm_count.times{
			if y > 0
				y -= arm_len
				w << 'U'
			else
				y += arm_len
				w << 'D'
			end
			arm_len /= 2
		}
		
#		puts [x, y]
		puts w
		
#		ax, ay = check_arms(arms, w)
		
#		p [ax, org_x]
#		p [ay, org_y]
		
	}
	
	
else
	puts "-1"
end