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# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Linq;
using System.Collections.Generic;
class M{static void Main(string[] _) {
  var N = int.Parse(Console.ReadLine());
  
  //////////////////////////////////////////  
  var L = new List<P>();
  for( int I=0 ; I<N ; ++I ) {
    var tmp = Console.ReadLine().Split(' ');
    L.Add( new P( int.Parse(tmp[0]) , int.Parse(tmp[1]) ) );
  }

  //////////////////////////////////////////  
  var L2 = new List<int>();
  var L4 = new List<int>();
  foreach( var A in L ) {
    L2.Add( (int)Math.Abs(A.x) ) ;
    L2.Add( (int)Math.Abs(A.y) ) ;
    int x2 = (int)Math.Abs(A.x) + (int)Math.Abs(A.y);
    L2.Add(x2);    
    L4.Add(x2);
  }
  L2.Sort( (a,b) => a - b );
  L4.Sort( (a,b) => a - b );  
  
  //////////////////////////////////////////  
  var L3 = new List<int>();
  int l0 = 0;
  foreach( var A in L2 ) {
    int B = A - l0;
    if ( 0 < B ) {
      L3.Add( B );
      l0 += B;
    }
  }
  L3.Sort( (a,b) => b - a );
  if ( L3.Count() <= 38 ) {
    L3.Add(1);
    L3.Add(1);
  }
  
  
  
  //////////////////////////////////////////  
  string result = "";
  foreach( var A in L ) {
    
    int x = A.x;
    int y = A.y;
    for( int I=0; I<L3.Count() ; ++ I ) {
      int l = L3[I];
      
      if ( Math.Abs(y) <= Math.Abs(x) ) {
        if ( 0 < x )   { result += "R"; x -= l; continue; }
        if ( x < 0 )   { result += "L"; x += l; continue; }
        if ( 0 < y )   { result += "U"; y -= l; continue; }
        if ( y < 0 )   { result += "D"; y += l; continue; }
      } else {
        if ( 0 < y )   { result += "U"; y -= l; continue; }
        if ( y < 0 )   { result += "D"; y += l; continue; }
        if ( 0 < x )   { result += "R"; x -= l; continue; }
        if ( x < 0 )   { result += "L"; x += l; continue; }
      }
      
      result += "R"; x -= l;
    }

    if ( x == 0 && y == 0 ) {
      result += "\n";
    } else {
  	  Console.WriteLine("-1");  
      return;
    }
    
  }
	
  Console.WriteLine( L3.Count() + "\n" + String.Join(" ",L3) );  
  Console.WriteLine(result);  
  
}}



class P {
  public int x;
  public int y;
  public P( int _x , int _y ) {x=_x;y=_y;}
  public int L() { return (int)( Math.Abs(x) + Math.Abs(y) ); }
  public bool isOdd() { return L() % 2 != 0 ; }
}