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: Accepted)

#include <bits/stdc++.h>
using namespace std;
#define rep(i, n) for(int i = 0; i < (n); ++i)

typedef long long ll;
typedef pair<ll, ll> P;

class dis{
    public:
        int x, y;

    int getDis(){ return abs(x) + abs(y); }
};

const int MAX = 1010;
const int INF = 1e9;
const int MOD = 1e9 + 7;

int N;
vector<dis> D;

int main(){
    cin >> N;
    rep(i, N){
        int x, y;
        cin >> x >> y;
        dis d;
        d.x = x;
        d.y = y;
        D.push_back(d);
    }

    int maxi = 0;
    rep(i, N - 1){
        int a = D[i].getDis();
        int b = D[i + 1].getDis();
        if(a % 2 != b % 2){
            cout << -1 << endl;
            return 0;
        }
        maxi = max(maxi, a);
    }
    maxi = max(maxi, D[N - 1].getDis());
    
    vector<ll> m;
    ll sum = 0;
    if(maxi % 2 == 0){
        m.push_back(1);
        sum++;
    }
    for(ll i = 1; sum < maxi; i*=2){
        m.push_back(i);
        sum += i;
    }

    sort(m.begin(), m.end(), greater<ll>());
    cout << m.size() << endl;
    for(auto i : m) cout << i << " ";
    cout << endl;
    rep(i, N){
        dis d = D[i];
        string s;
        ll res = sum;
        rep(j, m.size()){
            res -= m[j];
            ll nx, ny;

            nx = d.x + m[j];
            ny = d.y;
            if(abs(nx) + abs(ny) <= res){
                s.push_back('L');
                d.x = nx;
                d.y = ny;
                continue;
            }

            nx = d.x - m[j];
            if(abs(nx) + abs(ny) <= res){
                s.push_back('R');
                d.x = nx;
                d.y = ny;
                continue;
            }

            nx = d.x;
            ny = d.y + m[j];
            if(abs(nx) + abs(ny) <= res){
                s.push_back('D');
                d.x = nx;
                d.y = ny;
                continue;
            }

            ny = d.y - m[j];
            if(abs(nx) + abs(ny) <= res){
                s.push_back('U');
                d.x = nx;
                d.y = ny;
                continue;
            }
        }
        cout << s << endl;
    }
}