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 <iostream>
#include <vector>
#include <queue>
#include <algorithm>
#include <map>
#include <unordered_map>
#include <cassert>

using namespace std;

#define rep(i, n) for(int i = 0; i < n; ++i)
#define all(s) s.begin(), s.end()

int main() {
  int n;
  cin >> n;

  bool odd = false, even = false;
  vector< pair<int64_t, int64_t> > a(n);
  rep(i, n) {
    int64_t x, y;
    cin >> x >> y;

    // For Debug
    // cout << "x: " << x << ", y: " << y << endl;
    if ((x+y) & 1) {
      odd = true;
      // cout << "odd" << endl;
    } else {
      even = true;
      // cout << "even" << endl;
    }
    a[i].first = x;
    a[i].second = y;
  }
  if (odd && even) {
    cout << -1 << endl;
    return 0;
  }

  int m = 31;
  // int m = 3;
  cout << (even ? m+1 : m) << endl;

  vector<int64_t> d(m);
  rep(i, m) {
    d[i] = 1ll << i;
  }
  reverse(all(d));
  for (auto& x : d) {
    cout << x << ' ';
  }
  if (even) {
    cout << 1 << endl;
  } else {
    cout << endl;
  }

  for (auto& p : a) {
    int64_t x, y;
    tie(x, y) = p;
    if (even) { --x; }
    string res;
    rep(i, m) {
      bool u = x+y>0, v = x-y>0;
      if (u  && v)  { res += "R", x -= d[i]; continue; }
      if (!u && v)  { res += "D", y += d[i]; continue; }
      if (u  && !v) { res += "U", y -= d[i]; continue; }
      if (!u && !v) { res += "L", x += d[i]; continue; }
    }
    assert(x == 0 && y == 0);
    if (even) { res += "R"; };
    cout << res << endl;
  }
}