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)



#if 1
#include <iostream>
#include <fstream>
#include <string>
#include <vector>
#include <map>
#include <set>
#include <unordered_map>
#include <unordered_set>
#include <queue>
#include <stack>
#include <array>
#include <deque>
#include <algorithm>
#include <utility>
#include <cstdint>
#include <functional>
#include <iomanip>
#include <numeric>
#include <assert.h>
#include <bitset>
#include <list>

auto& in = std::cin;
auto& out = std::cout;
#define all_range(C) std::begin(C), std::end(C)
const double PI = 3.141592653589793238462643383279502884197169399375105820974944;

int32_t N;
int32_t x[1000];
int32_t y[1000];

int main()
{
	using std::endl;
	in.sync_with_stdio(false);
	out.sync_with_stdio(false);
	in.tie(nullptr);
	out.tie(nullptr);

	in >> N;
	for (size_t i = 0; i < N; i++)
	{
		in >> x[i]>>y[i];
		if (x[i] > 100 || y[i] > 100) { return 0; }
	}
	if (N == 1) {
		out << 2 << endl;
		out << abs(x[0]) << ' ' << abs(y[0]) << endl;
		out << "RL"[(x[0] >= 0)];
		out << "UD"[(y[0] >= 0)];
		return 0;
	}
	auto diff = abs(abs(x[0]) + abs(y[0]) - (abs(x[1]) + abs(y[1])));
	if (N >= 2)
	{
		for (size_t i = 2; i < N; i++)
		{
			if (abs(abs(abs(x[0]) + abs(y[0])) - (abs(x[i]) + abs(y[i]))) % 2 != diff%2) {
				out << -1 << endl;
				return 0;
			}
		}
	}

	int num = 0;
	if ((abs(x[0]) + abs(y[0])) % 2 == 0) {
		num = 30;
		out << 30 << "\n1 1";
	}
	else {
		num = 29;
		out << 29 << "\n1";
	}
	for (uint32_t tmp = 2; tmp < (1<<15); tmp*=2)
	{
		out << (char)' ' << 1;
		out << (char)' ' << 1;
	}
	out << '\n';

	for (size_t QUERY = 0; QUERY < N; QUERY++)
	{
		int tmp = 0;
		auto xx = x[QUERY];
		auto yy = (y[QUERY]);

		while (xx > 0) {
			out << (char)'R'; --xx; ++tmp;
		}
		while (xx < 0) {
			out << (char)'L'; ++xx; ++tmp;
		}

		while (yy > 0) {
			out << (char) 'U'; --yy; ++tmp;
		}
		while (yy < 0) {
			out << (char)'D'; ++yy; ++tmp;
		}

		while (tmp < num) {
			out << "UD";
			tmp += 2;
		}
		assert(tmp == num);
		out << '\n';
	}

	return 0;
}
#endif