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 <bits/stdc++.h>
using namespace std;

#define INF_LL (int64)1e18
#define INF (int32)1e9
#define REP(i, n) for(int64 i = 0;i < (n);i++)
#define FOR(i, a, b) for(int64 i = (a);i < (b);i++)
#define all(x) x.begin(),x.end()
#define fs first
#define sc second

using int32 = int_fast32_t;
using uint32 = uint_fast32_t;
using int64 = int_fast64_t;
using uint64 = uint_fast64_t;
using PII = pair<int32, int32>;
using PLL = pair<int64, int64>;

const double eps = 1e-10;

template<typename A, typename B>inline void chmin(A &a, B b){if(a > b) a = b;}
template<typename A, typename B>inline void chmax(A &a, B b){if(a < b) a = b;}

template<typename T>
vector<T> make_v(size_t a){return vector<T>(a);}

template<typename T,typename... Ts>
auto make_v(size_t a,Ts... ts){
  return vector<decltype(make_v<T>(ts...))>(a,make_v<T>(ts...));
}

template<typename T,typename U,typename... V>
typename enable_if<is_same<T, U>::value!=0>::type
fill_v(U &u,const V... v){u=U(v...);}

template<typename T,typename U,typename... V>
typename enable_if<is_same<T, U>::value==0>::type
fill_v(U &u,const V... v){
  for(auto &e:u) fill_v<T>(e,v...);
}

char dir[5] = "RUDL";
int64 rid[256]={};
int64 dx[4] = {1, -1, 1, -1};
int64 dy[4] = {1, 1, -1, -1};


vector<int64> calc(int64 x) {
	int64 xx = abs(x);
	int64 sgn = x < 0 ? 1 : 0;
	vector<int64> v;
	int64 bef = 0;
	for (int64 i = 38; i >= 0; i--) {
		if (bef) {
			v.push_back(1-sgn);
		} else {
			v.push_back(sgn);
		}
		if (xx >> i & 1) {
			bef = 0;
		} else {
			bef = 1;
		}
	}
	return v;
}

PLL topoint(const vector<int64>& d, string& s) {
	PLL p(0, 0);
	REP(i, s.size()) {
		if (s[i] == 'L') p.fs -= d[i];
		if (s[i] == 'R') p.fs += d[i];
		if (s[i] == 'U') p.sc += d[i];
		if (s[i] == 'D') p.sc -= d[i];
	}
	return p;
}

int main(void) {
	int64 N;
	cin >> N;
	vector<int64> X(N), Y(N);
	vector<int64> U(N), V(N);
	int64 parity = -1;
	REP(i, N) {
		cin >> X[i] >> Y[i];
		if (parity != -1 && parity != abs(X[i]+Y[i])%2) {
			cout << -1 << endl;
			return 0;
		} else {
			parity = abs(X[i]+Y[i])%2;
		}
		U[i] = X[i]-Y[i]; V[i] = X[i]+Y[i];
	}
	vector<int64> d(39, 0);
	string res = "";
	REP(i, 39) d[i] = 1LL << i;
	if (parity == 0) {
		d.push_back(1);
		res = "D";
		REP(i, N) {
			U[i] += -1;
			V[i] += 1;
		}
	}
	cout << d.size() << endl;
	REP(i, d.size()) cout << d[i] << " ";
	cout << endl;
	REP(i, N) {
		string ret = res;
		auto du = calc(U[i]);
		auto dv = calc(V[i]);
		REP(j, 39) {
			ret = dir[du[j] | (dv[j] << 1)] + ret;
		}
		// cout << U[i] << " " << V[i] << endl;
		// for (int64 j = 32; j >= 0; j--) {
		// 	cout << "(" << du[j] << " " << dv[j] << ") ";
		// }
		// cout << endl;
		// if (PLL(x, y) != topoint(d, ret)) {
		// 	cout << x << " " << y << endl;
		// }
		cout << ret << endl;
	}
}