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:
def robot_arms(N: int, XY: list) -> tuple:
    odd, even = 0, 0
    max_dist = 0
    for x, y in XY:
        ax, ay = abs(x), abs(y)
        if (ax + ay) % 2 == 0:
            even += 1
        else:
            odd += 1

        if max_dist < ax + ay:
            max_dist = ax + ay

    if odd == 0:
        # 偶数のみの場合は長さ 1 のアームが余計に必要
        arms = [1]
    elif even == 0:
        # 奇数のみの場合は 2 のべき乗の長さのアームだけで OK
        arms = []
    else:
        # 偶奇ごちゃ混ぜの場合は不成立
        return [], []

    # アームの準備
    tmp, sum_tmp = 1, sum(arms)
    while sum_tmp < max_dist:
        arms.append(tmp)
        sum_tmp += tmp
        tmp <<= 1

    # 各点へ腕を伸ばすための操作方法を決めていく
    ops = []
    for x, y in XY:
        op = ''
        # 決定操作
        # 長さが 2^m までの腕を使って届く範囲は綺麗な菱形になっている。
        # 点(x,y)から原点に向かって考える。
        # 腕の長い物から使っていき、使っていない腕で届く範囲に入っていく様に
        # 腕の方向を決定していく。
        # ただし、逆順に決定していくので、決定過程での方向と実際に出力すべき
        # 方向は逆になっていることに注意
        left_len = sum(arms)
        nx, ny = x, y
        for arm in reversed(arms):
            left_len -= arm
            if abs(nx + arm) + abs(ny) <= left_len:
                op = 'L' + op
                nx, ny = nx + arm, ny
            elif abs(nx - arm) + abs(ny) <= left_len:
                op = 'R' + op
                nx, ny = nx - arm, ny
            elif abs(nx) + abs(ny + arm) <= left_len:
                op = 'D' + op
                nx, ny = nx, ny + arm
            elif abs(nx) + abs(ny - arm) <= left_len:
                op = 'U' + op
                nx, ny = nx, ny - arm
        ops.append(op)

    return arms, ops


if __name__ == "__main__":
    N = int(input())
    XY = [tuple(map(int, input().split())) for _ in range(N)]
    ans = robot_arms(N, XY)

    if len(ans[0]) == 0:
        print(-1)
    else:
        d, w = ans
        print(len(d))
        print(' '.join(map(str, d)))
        for ww in w:
            print(ww)