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 (Clang 3.8.0) (judge verdict: Wrong Answer)

#include<cstdio>
#include<cstdlib>
using namespace std;
typedef long long int ll;

// definition {{{ 1

// scaning {{{ 2
#define Scd(x) scanf("%d", &x)
#define Scd2(x,y) scanf("%d%d", &x, &y)
#define Scd3(x,y,z) scanf("%d%d%d", &x, &y, &z)

#define Scll(x) scanf("%llu", &x)
#define Scll2(x,y) scanf("%llu%llu", &x, &y)
#define Scll3(x,y,z) scanf("%llu%llu%llu", &x, &y, &z)

#define Scc(c) scanf("%c", &c);
#define Scs(s) scanf("%s", s);
#define Scstr(s) scanf("%s", &s);
// }}} 2

// constants {{{ 2
#define EPS (1e-7)
#define INF (1e9)
#define PI (acos(-1))
// }}} 2

// systems {{{ 2
#define Rep(x,y) for(int x = 0; x < y; x++)
#define Repe(x,y,z) for(int x = z; x < y; x++)
// }}} 2

// output {{{ 2
#define YesNo(a) (a)?printf("Yes\n"):printf("No\n");
// }}} 2

// }}} 1

int main() {

    int n;
    ll x[1000],y[1000];
    ll leng[1000];

    Scd(n);

    Rep(i,n){
        Scd2(x[i],y[i]);
        leng[i] = abs(x[i]) + abs(y[i]);
    }

    int odd = leng[0] % 2;

    Rep(i,n){
        if(leng[i] % 2 != odd){
            printf ("%d\n", -1);
            return 0;
        }
    }

    int ans = 31;
    ans += !odd;
    printf ("%d\n", ans);

    ll longer[32] = {1};
    Rep(i,ans){
        printf("%lld", longer[i]);
        longer[i+1] = longer[i];
        longer[i+1] *= 2;
        if( i+1 != ans) printf (" ");
    }
    if(!odd) printf (" %d", 1);
    printf ("\n");

    ll pow31 = 1;
    Rep(i,31) pow31*=2;

    ll u[1000],v[1000];
    Rep(i,n){
        x[i] -= !odd;
        u[i] = x[i] + y[i];
        v[i] = x[i] - y[i];
        u[i] = (u[i] + (pow31 - 1)) / 2;
        v[i] = (v[i] + (pow31 - 1)) / 2;
        ll xx = 0;
        ll yy = 0;
        Rep(j,ans){
            if( !odd && j == ans-1) {printf ("R"); xx++ ;continue;}
            if( u[i] % 2 && v[i] % 2){
                printf ("R");
                xx += longer[j];
            }else if( u[i] % 2 == 0 && v[i] % 2){
                printf ("D");
                yy -= longer[j];
            }else if( u[i] % 2 && v[i] % 2 == 0 ){
                printf ("U");
                yy += longer[j];
            }else{
                printf ("L");
                xx -= longer[j];
            }
            u[i] >>= 1;
            v[i] >>= 1;
        }
        //printf ("%lld,%lld", xx,yy);
        printf ("\n");
    }

    return 0;
}