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 Python (3.4.3) (judge verdict: Wrong Answer)

n = int(input())
a = [list(map(int, input().split())) for i in range(n)]


ans_list = []

if sum(a[0]) % 2 == 1:
    for z in range(n):
        ans = []
        flag = 1
        x, y = a[z]
        rev_x, rev_y = False, False
        if x < 0:
            rev_x = True
            x = -x

        if y < 0:
            rev_y = True
            y = -y

        n = x + y
        move = [2**i for i in range(n.bit_length())]
        if n == 0:
            minus = 0
        else:
            minus = int("1"*(n.bit_length()), 2) - n

        if minus % 2 == 0:
            minus //= 2
        else:
            flag = 0

        minus = format(minus, "b")
        minus = minus[::-1]
        for i in range(len(minus)):
            if minus[i] == "1":
                move[i] *= -1

        for i in range(len(move)):
            if x % 2 ^ y % 2:
                if x % 2 == 1:
                    x -= move[i]//2**i
                    if move[i] > 0:
                        ans.append("R")
                    else:
                        ans.append("L")
                else:
                    y -= move[i]//2**i
                    if move[i] > 0:
                        ans.append("U")
                    else:
                        ans.append("D")
                x //= 2
                y //= 2

            else:
                flag = 0
                break

        if rev_x:
            for i in range(len(ans)):
                if ans[i] == "R":
                    ans[i] = "L"
                elif ans[i] == "L":
                    ans[i] = "R"
        if rev_y:
            for i in range(len(ans)):
                if ans[i] == "U":
                    ans[i] = "D"
                elif ans[i] == "D":
                    ans[i] = "U"

        if flag == 0:
            ans_list = [-1]
            break
        ans_list.append("".join(ans + ["R"*(40-len(ans))]))

    if ans_list == [-1]:
        print(-1)
    else:

        print(40)
        print(" ".join(map(str, [2**i for i in range(40)])))
        for i in ans_list:
            print(i)


else:
    for z in range(n):
        ans = ["R"]
        flag = 1
        x, y = a[z]
        x += 1
        rev_x, rev_y = False, False
        if x < 0:
            rev_x = True
            x = -x

        if y < 0:
            rev_y = True
            y = -y

        n = x + y
        move = [2**i for i in range(n.bit_length())]
        if n == 0:
            minus = 0
        else:
            minus = int("1"*(n.bit_length()), 2) - n

        if minus % 2 == 0:
            minus //= 2
        else:
            flag = 0

        minus = format(minus, "b")
        minus = minus[::-1]
        for i in range(len(minus)):
            if minus[i] == "1":
                move[i] *= -1

        for i in range(len(move)):
            if x % 2 ^ y % 2:
                if x % 2 == 1:
                    x -= move[i]//2**i
                    if move[i] > 0:
                        ans.append("R")
                    else:
                        ans.append("L")
                else:
                    y -= move[i]//2**i
                    if move[i] > 0:
                        ans.append("U")
                    else:
                        ans.append("D")
                x //= 2
                y //= 2

            else:
                flag = 0
                break

        if rev_x:
            for i in range(len(ans)):
                if ans[i] == "R":
                    ans[i] = "L"
                elif ans[i] == "L":
                    ans[i] = "R"
        if rev_y:
            for i in range(len(ans)):
                if ans[i] == "U":
                    ans[i] = "D"
                elif ans[i] == "D":
                    ans[i] = "U"

        if flag == 0:
            ans_list = [-1]
            break
        ans_list.append("".join(ans + ["R"*(40-len(ans))]))

    if ans_list == [-1]:
        print(-1)
    else:

        print(40)
        print(" ".join(map(str, [1] + [2**i for i in range(39)])))
        for i in ans_list:
            print(i)