Problem: 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

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.

Examples

Input

3
-1 0
0 3
2 -1


Output

2
1 2
RL
UU
DR


Input

5
0 0
1 0
2 0
3 0
4 0


Output

-1


Input

2
1 1
1 1


Output

2
1 1
RU
UR


Input

3
-7 -3
7 3
-3 -7


Output

5
3 1 4 1 5
LRDUL
RDULR
DULRD

Solution (in PYTHON3):

Solution:
import sys
fin = sys.stdin.readline

N = int(fin())
coords = [[int(elem) for elem in fin().split(' ')] for _ in range(N)]
odd_or_even = abs((coords[0][0] + coords[0][1]) % 2)
max_sum = 0
for coord in coords:
    x, y = coord
    if (x + y) % 2 != odd_or_even:
        print(-1)
        exit(0)
    else:
        max_sum = max(max_sum, abs(x) + abs(y))

# partial solution (because m must not exceed 40)
def partial_solution(max_sum, coords):
    m = max_sum
    print(m)
    print(' '.join(['1'] * m))
    for coord in coords:
        x, y = coord
        count = 0
        output = ""
        if x > 0:
            output += "R" * x
        else:
            output += "L" * abs(x)
        count += abs(x)

        if y > 0:
            output += "U" * y
        else:
            output += "D" * y
        count += abs(y)

        assert (m - count) % 2 == 0
        output += "LR" * ((m - count) // 2)
        print(output)
#partial_solution(max_sum, coords)

def padding(binary_expression):
    len_pad = 31 - len(binary_expression[2:])
    return "0" * len_pad + binary_expression[2:]


def full_solution(max_sum, coords, odd_or_even):
    mode_map = {(0, 0): "L",
                (1, 1): "R",
                (0, 1): "D",
                (1, 0): "U"}
    rotated_coords = [(x + y, x - y) for x, y in coords]
    d = []
    for i in range(31):
        d.append(2**i)  
    if odd_or_even == 1:
        m = 31
    else:
        m = 32
        d.append(1)
    print(m)
    print(" ".join([str(d_elem) for d_elem in d]))
    if odd_or_even == 1:
        for rotated_coord in rotated_coords:
            u, v = rotated_coord
            const_2pow31 = 2**31
            flags_u = list(padding(bin((u + const_2pow31 - 1) // 2)))[::-1]
            flags_v = list(padding(bin((v + const_2pow31 - 1) // 2)))[::-1]
            flags = [(int(fu), int(fv)) for fu, fv in zip(flags_u, flags_v)]
            output = "".join([mode_map[flag] for flag in flags])
            print(output)
    else:
        for rotated_coord in rotated_coords:
            u, v = rotated_coord
            u -= 1
            v -= 1
            const_2pow31 = 2**31
            flags_u = list(padding(bin((u + const_2pow31 - 1) // 2)))[::-1]
            flags_v = list(padding(bin((v + const_2pow31 - 1) // 2)))[::-1]
            flags = [(int(fu), int(fv)) for fu, fv in zip(flags_u, flags_v)]
            output = "".join([mode_map[flag] for flag in flags]) + "R"
            print(output)

full_solution(max_sum, coords, odd_or_even)