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 <algorithm>
#include <cmath>
#include <functional>
#include <iostream>
#include <string>
#include <vector>
#include <cassert>
#define rep(i, n) for(int i = 0; i < (int)(n); i++)
#define repR(i, n) for(int i = (n) - 1; i > -1; i--)
#define rep1(i, n) for(int i = 1; i < (int)(n + 1); i++)
#define rep1R(i, n) for(int i = (n); i > 0; i--)
typedef long long ll;
typedef unsigned long long ull;
using namespace std;

int main() {
  int N;
  cin >> N;
  vector<int> X(N), Y(N);
  rep(i, N) cin >> X[i] >> Y[i];
  bool all_even = true;
  bool all_odd = true;
  rep(i, N) {
    if ((abs(X[i]) + abs(Y[i])) % 2 == 0) all_odd = false;
    else all_even = false;
  }
  if (!all_even && !all_odd) {
    cout << -1 << endl;
    return 0;
  }
  int max_dis = 0;
  rep(i, N) {
    max_dis = max(max_dis, abs(X[i]) + abs(Y[i]));
  }
  int m = 32;
  if (all_even) m++;
  cout << m << endl;
  repR(i, m) {
    cout << (ll)pow(2, i) << " ";
  }
  if (all_even) m--;
  cout << endl;
  int c = 0;
  rep(i, N) {
    ll tmp_x = X[i], tmp_y = Y[i];
    if (all_even) {
      tmp_x -= 1;
      cout << "R";
    }
    repR(j, m) {
      int x_d = abs(tmp_x), y_d = abs(tmp_y);
      if (x_d > y_d) {
        if (tmp_x > 0) {
          tmp_x -= (ll)pow(2, j);
          cout << "R";
        } else {
          tmp_x += (ll)pow(2, j);
          cout << "L";
        }
      } else {
        if (tmp_y > 0) {
          tmp_y -= (ll)pow(2, j);
          cout << "U";
        } else {
          tmp_y += (ll)pow(2, j);
          cout << "D";
        }
      }
    }
    assert(tmp_x == 0 && tmp_y == 0);
    cout << endl;
  }
  return 0;
}