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 d_robot_arms(N, Pos):
    # 参考: http://drken1215.hatenablog.com/entry/2018/09/30/002900

    pos = []
    # すべての座標でパリティが一致しなければ、条件を満たす腕を選べない
    parity = (abs(Pos[0][0]) + abs(Pos[0][1])) % 2
    for x, y in Pos:
        parity_current = (abs(x) + abs(y)) % 2
        if parity != parity_current:
            return -1
        pos.append([x - y, x + y])
        # u=x-y, v=x+y とすると、 点(u, v)は
        # 原点周りに45度回転させる座標変換によって点(x, y)から移される点

    # log2(10**9)=29.90 なので、2**30以下の2の冪乗を適当に選べば
    # xy平面上の x+y が奇数である任意の座標に到達できる
    arms = [1 << k for k in range(30, -1, -1)]
    if parity % 2 == 0:
        arms.append(1)  # x+yが偶数である座標に到達するときに必要

    # 腕の数, 長さのリストを最初に格納する(任意の入力に対して固定)
    ans_tmp = [str(len(arms)), ' '.join(list(map(str, arms)))]

    for pos_u, pos_v in pos:
        # u, v座標はそれぞれ独立に見て構わない
        usum, vsum = 0, 0
        tmp = ''
        for arm_len in arms:
            if usum <= pos_u:
                udir = 'positive'
                usum += arm_len
            else:
                udir = 'negative'
                usum -= arm_len

            if vsum <= pos_v:
                vdir = 'positive'
                vsum += arm_len
            else:
                vdir = 'negative'
                vsum -= arm_len

            # uv平面で軸と平行にどちら向きに移動するかと
            # xy平面での移動モード(UDLR)との対応
            if udir == 'positive' and vdir == 'positive':
                tmp += 'R'
            elif udir == 'positive' and vdir == 'negative':
                tmp += 'D'
            elif udir == 'negative' and vdir == 'negative':
                tmp += 'L'
            else:
                tmp += 'U'
        ans_tmp.append(''.join(tmp))
    ans = '\n'.join(ans_tmp)
    return ans

N = int(input())
Pos = [[int(i) for i in input().split()] for j in range(N)]
print(d_robot_arms(N, Pos))