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 C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
#define SORT(x) sort((x).begin(), (x).end())
#define ALL(x) (x).begin(), (x).end()
#define rep(i, n) for (ll i = 0; i < n; i++)
#define reps(i, m, n) for (ll i = m; i < n; i++)
#define repr(i, m, n) for (ll i = m; i >= n; i--)
#define de(x) cout << #x << "=" << x << endl;
template <class T>
bool maxi(T &a, const T &b) {
    if (a < b) {
        a = b;
        return 1;
    }
    return 0;
}
template <class T>
bool mini(T &a, const T &b) {
    if (b < a) {
        a = b;
        return 1;
    }
    return 0;
}
#define INF2 1000000000000000037
#define INF 1000000007
#define MOD 1000000007
using namespace std;
using ll = long long;
using P = pair<ll, ll>;
//--GLOBAL---------------------------------



//--MAIN-----------------------------------
void Main() {
    ll N; cin>>N;
    vector<ll> X(N);
    vector<ll> Y(N);
    vector<ll> U(N);
    vector<ll> V(N);
    rep(i,N){ cin>>X[i]; cin>>Y[i]; U[i]=X[i]+Y[i]; V[i]=X[i]-Y[i];}

    bool kisu;

    rep(i,N){
        if(i==0){
            kisu = U[i]%2;
        }
        else{
            if( kisu!= U[i]%2 ){
                cout<<  -1 <<"\n"; return;
            }
        }
        if(!kisu){
            X[i]+=1;
            U[i]+=1;
            V[i]+=1;
        }
    }

    ll m=33;
    //ll m=7;
    ll d[m+1];
    string ans[N]={};

    rep(i,m){
        d[i]=pow(2,(m-1-i));
    }

    ll nowU[N]={};
    ll nowV[N]={};

    rep(j,m){
        rep(i,N){
            bool Uplus,Vplus;
            if(abs(U[i]-(nowU[i]+d[j]))<=abs(U[i]-(nowU[i]-d[j]))){
                Uplus=true; nowU[i]+=d[j];
            }
            else{
                Uplus=false; nowU[i]-=d[j];
            }
            if(abs(V[i]-(nowV[i]+d[j]))<=abs(V[i]-(nowV[i]-d[j]))){
                Vplus=true; nowV[i]+=d[j];
            }
            else{
                Vplus=false; nowV[i]-=d[j];
            }
            if(Uplus and Vplus) ans[i]+="R";
            if(Uplus and !Vplus) ans[i]+="U";
            if(!Uplus and Vplus) ans[i]+="D";
            if(!Uplus and !Vplus) ans[i]+="L";
            /*
            de(i);
            de(j);
            de(nowU[i]);
            de(nowV[i]);
            */
        }
    }

    if(!kisu){
        ++m;
        d[m-1]=1;
        rep(i,N){
            ans[i]+="L";
        }
    }

    cout<< m<<"\n";
    rep(i,m){
        cout<< d[i] <<" ";
    }
    cout<<"\n";
    rep(i,N){
        cout<< ans[i]<<"\n";
    }

}   
//--START----------------------------------
int main() {
    cin.tie(nullptr);
    ios_base::sync_with_stdio(false);
    cout << fixed << setprecision(15);
    Main();
}
//-----------------------------------------