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>
typedef long long ll;
typedef long double ld;
const int INF=1e9,MOD=1e9+7,ohara=1e6;
const ll LINF=1e18;
using namespace std;

#define rep(i,n) for(int (i)=0;(i)<(int)(n);(i)++)
#define rrep(i,a,b) for(int i=(a);i<(b);i++)
#define rrrep(i,a,b) for(int i=(a);i>=(b);i--)
#define all(v) (v).begin(), (v).end()
#define Size(n) (n).size()
#define Cout(x) cout<<(x)<<endl

ll n,cnt,ans,a,b,c,d[ohara],tmp,tmpp,m,h,w,x[ohara],y[ohara],sum,pos,k;
ld doua;
int dy[]={1,0,-1,0};
int dx[]={0,1,0,-1};
//int dy[]={-1,0,1,-1,1,-1,0,1};
//int dx[]={-1,-1,-1,0,0,1,1,1};
string alph("abcdefghijklmnopqrstuvwxyz");
bool fl;
struct edge{int to,cost;};

//-------------------------↓↓↓↓↓↓------------------------

int check(){
    int ret=1;
    ll dis=abs(x[n-1])+abs(y[n-1]);
    rep(i,n){
        if((abs(x[i])+abs(y[i]))%2!=dis%2)ret=-1;
    }
    return ret;
}

int main(void){
       cin.tie(0);
    ios::sync_with_stdio(false);

    cin>>n;
    rep(i,n)cin>>x[i]>>y[i];
    if(check()==-1)Cout(-1);
    else{
        Cout(30);
        cnt=1;
        rep(i,30){
            d[i]=cnt;
            cnt*=2;
        }
        reverse(d,d+30);
        rep(i,30){
            if(i==29)Cout(d[i]);
            else cout<<d[i]<<" ";
        }
        rep(i,n){
            string s;
            rep(j,30){
                if(abs(x[i])<abs(y[i])){
                    if(y[i]<=0){
                        y[i]+=d[j];
                        s+='D';
                    }
                    else{
                        y[i]-=d[j];
                        s+='U';
                    }
                }
                else{
                    if(x[i]<=0){
                        x[i]+=d[j];
                        s+='L';
                    }
                    else{
                        x[i]-=d[j];
                        s+='R';
                    }
                }
            }
            Cout(s);
        }
    }
    return 0;
}