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 CPP):

Solution:
#include <iostream>
#include <vector>
#include <string>
#include <cassert>

using namespace std;
typedef long long ll;

void write_mode(const vector<ll> arms, ll x, ll y){
    string s;

    for(ll d : arms){
        if(x > 0 && abs(x) > abs(y)){
            s.push_back('R');
            x -= d;
        }else if(x < 0 && abs(x) > abs(y)){
            s.push_back('L');
            x += d;
        }else if(y > 0 && abs(x) < abs(y)){
            s.push_back('U');
            y -= d;
        }else{
            s.push_back('D');
            y += d;
        }
    }
    assert(x == 0 && y == 0);
    assert(s.size() == arms.size());
    cout << s << endl;
}

int main(){
    ll n;
    cin >> n;
    vector<pair<ll, ll>> p(n);
    for(int i = 0; i < n; ++i){
        cin >> p.at(i).first;
        cin >> p.at(i).second;
    }

    for(auto xy : p){
        if((abs(p.at(0).first) + abs(p.at(0).second)) % 2 !=
           (abs(xy.first) + abs(xy.second)) % 2){
            cout << -1 << endl;
            return 0;
        }
    }

    bool is_even = ((abs(p.at(0).first) + abs(p.at(0).second)) % 2 == 0);
    vector<ll> arms;
    if(is_even){
        arms.push_back(1);
    }
    for(int i = 30; i >= 0; --i){
        arms.push_back(1LL << i);
    }

    cout << arms.size() << endl;
    for(int i = 0; i < arms.size(); ++i){
        cout << arms.at(i);
        if(i != arms.size() - 1){
            cout << ' ';
        }else{
            cout << endl;
        }
    }

    for(auto xy : p){
        write_mode(arms, xy.first, xy.second);
    }

    return 0;
}