Problem:
Score : 2100 points

Problem Statement

Snuke has a board with an N \times N grid, and N \times N tiles.

Each side of a square that is part of the perimeter of the grid is attached with a socket.
That is, each side of the grid is attached with N sockets, for the total of 4 \times N sockets.
These sockets are labeled as follows:

The sockets on the top side of the grid: U1, U2, ..., UN from left to right

The sockets on the bottom side of the grid: D1, D2, ..., DN from left to right

The sockets on the left side of the grid: L1, L2, ..., LN from top to bottom

The sockets on the right side of the grid: R1, R2, ..., RN from top to bottom

Figure: The labels of the sockets

Snuke can insert a tile from each socket into the square on which the socket is attached.
When the square is already occupied by a tile, the occupying tile will be pushed into the next square, and when the next square is also occupied by another tile, that another occupying tile will be pushed as well, and so forth.
Snuke cannot insert a tile if it would result in a tile pushed out of the grid.
The behavior of tiles when a tile is inserted is demonstrated in detail at Sample Input/Output 1.

Snuke is trying to insert the N \times N tiles one by one from the sockets, to reach the state where every square contains a tile.
Here, he must insert exactly U_i tiles from socket Ui, D_i tiles from socket Di, L_i tiles from socket Li and R_i tiles from socket Ri.
Determine whether it is possible to insert the tiles under the restriction. If it is possible, in what order the tiles should be inserted from the sockets?

Constraints

1≦N≦300

U_i,D_i,L_i and R_i are non-negative integers.

The sum of all values U_i,D_i,L_i and R_i is equal to N \times N.

Partial Scores

2000 points will be awarded for passing the test set satisfying N≦40.

Additional 100 points will be awarded for passing the test set without additional constraints.

Input

The input is given from Standard Input in the following format:

N
U_1 U_2 ... U_N
D_1 D_2 ... D_N
L_1 L_2 ... L_N
R_1 R_2 ... R_N

Output

If it is possible to insert the tiles so that every square will contain a tile, print the labels of the sockets in the order the tiles should be inserted from them, one per line. If it is impossible, print NO instead. If there exists more than one solution, print any of those.

Sample Input 1

3
0 0 1
1 1 0
3 0 1
0 1 1

Sample Output 1

L1
L1
L1
L3
D1
R2
U3
R3
D2

Snuke can insert the tiles as shown in the figure below. An arrow indicates where a tile is inserted from, a circle represents a tile, and a number written in a circle indicates how many tiles are inserted before and including the tile.

Sample Input 2

2
2 0
2 0
0 0
0 0

Sample Output 2

NO

Solution in C (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <stdio.h>
#include <stdlib.h>

#define dag_valtype int
#define graph_valtype int

typedef struct graph_edge_sub graph_edge;

typedef struct {
	int num;
	int next_num;
	graph_edge *next;
	int prev_num;
}graph_vertex_sub;

struct graph_edge_sub{
	graph_vertex_sub *v;
	int w;
	graph_edge *next;
};

typedef struct graph_v_sub graph_vertex;

struct graph_v_sub{
	int num;
	graph_valtype val;
	int next_num;
	graph_vertex **next;
	int *next_weight;
	int prev_num;
	graph_vertex **prev;
	int *prev_weight;
};

typedef struct {
	int N;
	graph_vertex_sub **v_s;
	graph_vertex **v;
}graph;

//頂点数N, 各頂点の初期値ini_valのグラフを作る
graph *make_graph(int N, graph_valtype ini_val){
	int i;
	graph *g = (graph *)malloc(sizeof(graph));
	g->N = N;
	g->v_s = (graph_vertex_sub **)malloc(sizeof(graph_vertex_sub *) * N);
	g->v = (graph_vertex **)malloc(sizeof(graph_vertex *) * N);
	for(i = 0; i < N; i++){
		(g->v_s)[i] = (graph_vertex_sub *)malloc(sizeof(graph_vertex_sub));
		(g->v_s)[i]->num = i;
		(g->v_s)[i]->next_num = 0;
		(g->v_s)[i]->next = NULL;
		(g->v_s)[i]->prev_num = 0;
		(g->v)[i] = (graph_vertex *)malloc(sizeof(graph_vertex));
		(g->v)[i]->num = i;
		(g->v)[i]->val = ini_val;
		(g->v)[i]->next_num = 0;
		(g->v)[i]->next = NULL;
		(g->v)[i]->next_weight = NULL;
		(g->v)[i]->prev_num = 0;
		(g->v)[i]->prev = NULL;
		(g->v)[i]->prev_weight = NULL;
	}
	return g;
}

//グラフgの頂点aから頂点bに重みwの有向辺を張る (0 <= a, b <= N - 1)
void set_edge_graph(int a, int b, int w, graph *g){
	graph_edge *new1 = (graph_edge *)malloc(sizeof(graph_edge));
	new1->v = (g->v_s)[b];
	new1->w = w;
	new1->next = (g->v_s)[a]->next;
	(g->v_s)[a]->next = new1;
	(g->v_s)[a]->next_num++;
	(g->v_s)[b]->prev_num++;
}

//set_edge_graph後に呼び出す
void build_graph(graph *g){
	int i;
	graph_vertex_sub **v_s = g->v_s;
	graph_vertex **v = g->v;
	graph_vertex *nowv;
	graph_edge *nowe;
	for(i = 0; i < g->N; i++){
		v[i]->next = (graph_vertex **)malloc(sizeof(graph_vertex *) * v_s[i]->next_num);
		v[i]->next_weight = (int *)malloc(sizeof(int) * v_s[i]->next_num);
		v[i]->prev = (graph_vertex **)malloc(sizeof(graph_vertex *) * v_s[i]->prev_num);
		v[i]->prev_weight = (int *)malloc(sizeof(int) * v_s[i]->prev_num);
	}
	for(i = 0; i < g->N; i++){
		nowv = v[i];
		for(nowe = v_s[i]->next; nowe != NULL; nowe = nowe->next){
			(nowv->next)[nowv->next_num] = v[nowe->v->num];
			(nowv->next_weight)[nowv->next_num] = nowe->w;
			nowv->next_num++;
			(v[nowe->v->num]->prev)[v[nowe->v->num]->prev_num] = nowv;
			(v[nowe->v->num]->prev_weight)[v[nowe->v->num]->prev_num] = nowe->w;
			v[nowe->v->num]->prev_num++;
		}
	}
}

typedef struct SCC_sub{
	int num; //強連結成分番号
	int vertex_num; //強連結成分に含まれる頂点の個数
	int *vertexs; //強連結成分に含まれる元のグラフでの頂点番号
	dag_valtype val;
	int next_num;
	struct SCC_sub **next;
	int *next_weight;
	int prev_num;
	struct SCC_sub **prev;
	int *prev_weight;
}SCC; //強連結成分(Strongly Connected Components)

typedef struct {
	int N;
	SCC **sorted_SCC; //topological sort済み
}DAG; //非閉路有向グラフ(Directed Acyclic Graph)

int dfs(int next_bt, int *used, int *bt, int *bt_inv, graph_vertex *v){
	if(used[v->num] == 1){
		return next_bt;
	}
	else{
		int i;
		used[v->num] = 1;
		for(i = 0; i < v->next_num; i++){
			next_bt = dfs(next_bt, used, bt, bt_inv, v->next[i]);
		}
		bt[v->num] = next_bt;
		bt_inv[next_bt] = v->num;
		return next_bt + 1;
	}
}

void dfs_rev1(int next_SCC_num, int *used, int *SCC_num, int *next_num, int *prev_num, graph_vertex *v){
	if(used[v->num] == 1){
		int i;
		used[v->num] = 2;
		for(i = 0; i < v->prev_num; i++){
			dfs_rev1(next_SCC_num, used, SCC_num, next_num, prev_num, v->prev[i]);
		}
		SCC_num[v->num] = next_SCC_num;
		for(i = 0; i < v->next_num; i++){
			if(used[v->next[i]->num] == 1){
				next_num[v->num]++;
				prev_num[v->next[i]->num]++; //ここで自己ループを数えてしまう可能性がある
			}
		}
	}
}

void dfs_rev2(SCC *now_SCC, int *used, int *SCC_num, graph_vertex *v, DAG *dag){
	if(used[v->num] == 2){
		int i;
		SCC *next_SCC;
		used[v->num] = 3;
		for(i = 0; i < v->prev_num; i++){
			dfs_rev2(now_SCC, used, SCC_num, v->prev[i], dag);
		}
		now_SCC->vertexs[now_SCC->vertex_num] = v->num;
		now_SCC->vertex_num++;
		for(i = 0; i < v->next_num; i++){
			if(used[v->next[i]->num] == 2){
				next_SCC = dag->sorted_SCC[SCC_num[v->next[i]->num]];
				if(now_SCC->num != next_SCC->num){ //自己ループがないようにしている
					now_SCC->next[now_SCC->next_num] = next_SCC;
					now_SCC->next_weight[now_SCC->next_num] = v->next_weight[i];
					now_SCC->next_num++;
					next_SCC->prev[next_SCC->prev_num] = now_SCC;
					next_SCC->prev_weight[next_SCC->prev_num] = v->next_weight[i];
					next_SCC->prev_num++;
				}
			}
		}
	}
}

DAG *build_DAG(graph *g, dag_valtype ini_val){
	int N = g->N, i;
	int *used = (int *)malloc(sizeof(int) * N);

	int *bt = (int *)malloc(sizeof(int) * N);
	int *bt_inv = (int *)malloc(sizeof(int) * N);
	for(i = 0; i < N; i++){
		used[i] = 0;
		bt[i] = -1;
		bt_inv[i] = -1;
	}
	int next_bt;
	for(i = 0, next_bt = 0; i < N; i++){
		next_bt = dfs(next_bt, used, bt, bt_inv, g->v[i]);
	}

	int *SCC_num = (int *)malloc(sizeof(int) * N);
	int *next_num = (int *)malloc(sizeof(int) * N);
	int *prev_num = (int *)malloc(sizeof(int) * N);
	for(i = 0; i < N; i++){
		SCC_num[i] = 0;
		next_num[i] = 0;
		prev_num[i] = 0;
	}
	int next_SCC_num;
	for(i = N - 1, next_SCC_num = 0; i >= 0; i--){
		if(used[bt_inv[i]] == 1){
			dfs_rev1(next_SCC_num, used, SCC_num, next_num, prev_num, g->v[bt_inv[i]]);
			next_SCC_num++;
		}
	}

	DAG *dag = (DAG *)malloc(sizeof(DAG));
	SCC *now_SCC;
	dag->N = next_SCC_num;
	dag->sorted_SCC = (SCC **)malloc(sizeof(SCC *) * dag->N);
	for(i = 0; i < dag->N; i++){
		dag->sorted_SCC[i] = (SCC *)malloc(sizeof(SCC));
		now_SCC = dag->sorted_SCC[i];
		now_SCC->num = i;
		now_SCC->vertex_num = 0;
		now_SCC->vertexs = NULL;
		now_SCC->val = ini_val;
		now_SCC->next_num = 0;
		now_SCC->next = NULL;
		now_SCC->next_weight = NULL;
		now_SCC->prev_num = 0;
		now_SCC->prev = NULL;
		now_SCC->prev_weight = NULL;
	}
	for(i = 0; i < N; i++){
		now_SCC = dag->sorted_SCC[SCC_num[i]];
		now_SCC->vertex_num++;
		now_SCC->next_num += next_num[i];
		now_SCC->prev_num += prev_num[i];
	}
	for(i = 0; i < dag->N; i++){
		now_SCC = dag->sorted_SCC[i];
		now_SCC->vertexs = (int *)malloc(sizeof(int) * now_SCC->vertex_num);
		now_SCC->next = (SCC **)malloc(sizeof(SCC *) * now_SCC->next_num);
		now_SCC->next_weight = (int *)malloc(sizeof(int) * now_SCC->next_num);
		now_SCC->prev = (SCC **)malloc(sizeof(SCC *) * now_SCC->prev_num);
		now_SCC->prev_weight = (int *)malloc(sizeof(int) * now_SCC->prev_num);
		now_SCC->vertex_num = 0;
		now_SCC->next_num = 0;
		now_SCC->prev_num = 0;
	}

	for(i = N - 1, next_SCC_num = 0; i >= 0; i--){
		if(used[bt_inv[i]] == 2){
			dfs_rev2(dag->sorted_SCC[next_SCC_num], used, SCC_num, g->v[bt_inv[i]], dag);
			next_SCC_num++;
		}
	}

	return dag;
}

typedef struct {
	int i;
	int v;
}pair;

typedef struct {
	int x;
	int y;
}pos;

int pos_to_int(int x, int y, int N){
	return N * x + y;
}

pos int_to_pos(int num, int N){
	return (pos){num / N, num % N};
}

//v降順
signed compair(const void *a, const void *b){
	return ((pair *)b)->v - ((pair *)a)->v;
}

int search(int num, int SCC_num, int N, int **B2, graph *g, int **visited){
	pos p = int_to_pos(num, N);
	if(B2[p.x][p.y] != SCC_num || visited[p.x][p.y] == 2 || visited[p.x][p.y] == -1){
		return 0;
	}
	else if(visited[p.x][p.y] == 1){
		visited[p.x][p.y] = 2;
		return 1;
	}
	else{
		visited[p.x][p.y] = 1;
		graph_vertex *nowv = g->v[num];
		int i;
		for(i = 0; i < nowv->next_num; i++){
			if(search(nowv->next[i]->num, SCC_num, N, B2, g, visited) == 1){
				if(visited[p.x][p.y] == 2){
					return 0;
				}
				else{
					visited[p.x][p.y] = 2;
					return 1;
				}
			}
		}
		visited[p.x][p.y] = -1;
		return 0;
	}
}

int main(){
	int N, i, j, k;
	scanf("%d", &N);
	int *U = (int *)malloc(sizeof(int) * N);
	int *D = (int *)malloc(sizeof(int) * N);
	int *L = (int *)malloc(sizeof(int) * N);
	int *R = (int *)malloc(sizeof(int) * N);
	for(i = 0; i < N; i++){
		scanf("%d", &U[i]);
	}
	for(i = 0; i < N; i++){
		scanf("%d", &D[i]);
	}
	for(i = 0; i < N; i++){
		scanf("%d", &L[i]);
	}
	for(i = 0; i < N; i++){
		scanf("%d", &R[i]);
	}
	pair *UD = (pair *)malloc(sizeof(pair) * N);
	pair *LR = (pair *)malloc(sizeof(pair) * N);
	for(i = 0; i < N; i++){
		UD[i].i = i;
		UD[i].v = U[i] + D[i];
		LR[i].i = i;
		LR[i].v = N - (L[i] + R[i]);
	}
	qsort(UD, N, sizeof(pair), compair);
	qsort(LR, N, sizeof(pair), compair);
	int **B0 = (int **)malloc(sizeof(int *) * N);
	for(i = 0; i < N; i++){
		B0[i] = (int *)malloc(sizeof(int) * N);
		for(j = 0; j < N; j++){
			B0[i][j] = 0;
		}
	}
/*	printf("test0\n");
	for(i = 0; i < N; i++){
		printf("LR[%d] = (%d, %d)\n", i, LR[i].i, LR[i].v);
	}
	for(i = 0; i < N; i++){
		printf("UD[%d] = (%d, %d)\n", i, UD[i].i, UD[i].v);
	}
*/	int lastv, l, r, tmp;
	for(i = 0; i < N; i++){
//		printf("i = %d\n", i);
		if(LR[i].v == 0){
			break;
		}
		for(j = 0; j < LR[i].v; j++){
			B0[i][j] = 1;
		}
		lastv = UD[j - 1].v;
		for(l = 0; UD[l].v != lastv; l++);
		for(r = N - 1; UD[r].v != lastv; r--);
		for(k = 0; k <= (r - l) / 2; k++){
			tmp = B0[i][l + k];
			B0[i][l + k] = B0[i][r - k];
			B0[i][r - k] = tmp;
		}
		for(j = 0; j < N; j++){
			UD[j].v -= B0[i][j];
			if(UD[j].v < 0){
				printf("NO\n");
				return 0;
			}
		}
	}
/*	printf("B0\n");
	for(i = 0; i < N; i++){
		for(j = 0; j < N; j++){
			printf("%d", B0[i][j]);
		}
		printf("\n");
	}
*/	int **B1 = (int **)malloc(sizeof(int *) * N);
	int **B2 = (int **)malloc(sizeof(int *) * N);
	char **A = (char **)malloc(sizeof(char *) * N);
	int **visited = (int **)malloc(sizeof(int *) * N);
	for(i = 0; i < N; i++){
		B1[i] = (int *)malloc(sizeof(int) * N);
		B2[i] = (int *)malloc(sizeof(int) * N);
		A[i] = (char *)malloc(sizeof(char) * N);
		visited[i] = (int *)malloc(sizeof(int) * N);
	}
	for(i = 0; i < N; i++){
		for(j = 0; j < N; j++){
			B1[LR[i].i][UD[j].i] = B0[i][j];
		}
	}
	graph *g;
	DAG *dag;
	pos p;
	while(1){
/*		printf("B1:\n");
		for(i = 0; i < N; i++){
			for(j = 0; j < N; j++){
				printf("%d", B1[i][j]);
			}
			printf("\n");
		}
*/		for(i = 0; i < N; i++){
			for(j = 0, k = 0; k < L[i]; j++){
				if(B1[i][j] == 0){
					A[i][j] = 'L';
					k++;
				}
			}
			for(j = N - 1, k = 0; k < R[i]; j--){
				if(B1[i][j] == 0){
					A[i][j] = 'R';
					k++;
				}
			}
		}
		for(j = 0; j < N; j++){
			for(i = 0, k = 0; k < U[j]; i++){
				if(B1[i][j] == 1){
					A[i][j] = 'U';
					k++;
				}
			}
			for(i = N - 1, k = 0; k < D[j]; i--){
				if(B1[i][j] == 1){
					A[i][j] = 'D';
					k++;
				}
			}
		}
		g = make_graph(N * N, 0);
		for(i = 0; i < N; i++){
			for(j = 0; j < N; j++){
				if(B1[i][j] == 1){
					for(k = j + 1; k < N; k++){
						if(A[i][k] == 'L'){
							set_edge_graph(pos_to_int(i, j, N), pos_to_int(i, k, N), 1, g);
						}
					}
				}
			}
			for(j = N - 1; j >= 0; j--){
				if(B1[i][j] == 1){
					for(k = j - 1; k >= 0; k--){
						if(A[i][k] == 'R'){
							set_edge_graph(pos_to_int(i, j, N), pos_to_int(i, k, N), 1, g);
						}
					}
				}
			}
		}
		for(j = 0; j < N; j++){
			for(i = 0; i < N; i++){
				if(B1[i][j] == 0){
					for(k = i + 1; k < N; k++){
						if(A[k][j] == 'U'){
							set_edge_graph(pos_to_int(i, j, N), pos_to_int(k, j, N), 1, g);
						}
					}
				}
			}
			for(i = N - 1; i >= 0; i--){
				if(B1[i][j] == 0){
					for(k = i - 1; k >= 0; k--){
						if(A[k][j] == 'D'){
							set_edge_graph(pos_to_int(i, j, N), pos_to_int(k, j, N), 1, g);
						}
					}
				}
			}
		}
		build_graph(g);
/*		for(i = 0; i < g->N; i++){
			printf("i = %d: ", i);
			graph_vertex *nowv = g->v[i];
			for(j = 0; j < nowv->next_num; j++){
				printf("%d ", nowv->next[j]->num);
			}
			printf("\n");
		}
*/		dag = build_DAG(g, 0);
//		printf("dag->N = %d\n", dag->N);
		if(dag->N == N * N){
			break;
		}
		SCC *now_SCC;
		for(i = 0; i < dag->N; i++){
			now_SCC = dag->sorted_SCC[i];
			for(j = 0; j < now_SCC->vertex_num; j++){
				p = int_to_pos(now_SCC->vertexs[j], N);
				B2[p.x][p.y] = now_SCC->num;
			}
		}
		for(i = 0; i < N; i++){
			for(j = 0; j < N; j++){
				visited[i][j] = 0;
			}
		}
		for(i = 0; i < dag->N; i++){
			now_SCC = dag->sorted_SCC[i];
			if(now_SCC->vertex_num == 1){
				continue;
			}
			for(j = 0; j < now_SCC->vertex_num; j++){
				search(now_SCC->vertexs[j], now_SCC->num, N, B2, g, visited);
			}
		}
		for(i = 0; i < N; i++){
			for(j = 0; j < N; j++){
				if(visited[i][j] == 2){
					B1[i][j] ^= 1;
				}
			}
		}
		return 0;
	}
//	printf("break\n");
/*	printf("A:\n");
	for(i = 0; i < N; i++){
		for(j = 0; j < N; j++){
			printf("%c", A[i][j]);
		}
		printf("\n");
	}
*/	g = make_graph(N * N, 0);
	for(i = 0; i < N; i++){
		for(j = 0; j < N; j++){
			for(k = j + 1; k < N; k++){
				if(A[i][k] == 'L'){
					set_edge_graph(pos_to_int(i, j, N), pos_to_int(i, k, N), 1, g);
				}
			}
		}
		for(j = N - 1; j >= 0; j--){
			for(k = j - 1; k >= 0; k--){
				if(A[i][k] == 'R'){
					set_edge_graph(pos_to_int(i, j, N), pos_to_int(i, k, N), 1, g);
				}
			}
		}
	}
	for(j = 0; j < N; j++){
		for(i = 0; i < N; i++){
			for(k = i + 1; k < N; k++){
				if(A[k][j] == 'U'){
					set_edge_graph(pos_to_int(i, j, N), pos_to_int(k, j, N), 1, g);
				}
			}
		}
		for(i = N - 1; i >= 0; i--){
			for(k = i - 1; k >= 0; k--){
				if(A[k][j] == 'D'){
					set_edge_graph(pos_to_int(i, j, N), pos_to_int(k, j, N), 1, g);
				}
			}
		}
	}
	build_graph(g);
/*	for(i = 0; i < g->N; i++){
		printf("i = %d: ", i);
		graph_vertex *nowv = g->v[i];
		for(j = 0; j < nowv->next_num; j++){
			printf("%d ", nowv->next[j]->num);
		}
		printf("\n");
	}
*/	dag = build_DAG(g, 0);
//	printf("dag->N = %d\n", dag->N);
	char *ansc = (char *)malloc(sizeof(char) * N * N);
	int *ansd = (int *)malloc(sizeof(int) * N * N);
	for(i = 0; i < dag->N; i++){
		p = int_to_pos(dag->sorted_SCC[i]->vertexs[0], N);
		ansc[i] = A[p.x][p.y];
		if(B1[p.x][p.y] == 0){
			ansd[i] = p.x;
		}
		else{
			ansd[i] = p.y;
		}
	}
	//deback
/*	int **B3 = (int **)malloc(sizeof(int *) * N);
	for(i = 0; i < N; i++){
		B3[i] = (int *)malloc(sizeof(int) * N);
		for(j = 0; j < N; j++){
			B3[i][j] = -1;
		}
	}
	for(i = 0; i < N * N; i++){
		if(ansc[i] == 'L'){
			for(j = 0; j < N; j++){
				if(B3[ansd[i]][j] == -1){
					B3[ansd[i]][j] = i;
					break;
				}
			}
			if(j == N){
				printf("doubt!\n");
				return 0;
			}
		}
		if(ansc[i] == 'R'){
			for(j = N - 1; j >= 0; j--){
				if(B3[ansd[i]][j] == -1){
					B3[ansd[i]][j] = i;
					break;
				}
			}
			if(j == -1){
				printf("doubt!\n");
				return 0;
			}
		}
		if(ansc[i] == 'U'){
			for(j = 0; j < N; j++){
				if(B3[j][ansd[i]] == -1){
					B3[j][ansd[i]] = i;
					break;
				}
			}
			if(j == N){
				printf("doubt!\n");
				return 0;
			}
		}
		if(ansc[i] == 'D'){
			for(j = N - 1; j >= 0; j--){
				if(B3[j][ansd[i]] == -1){
					B3[j][ansd[i]] = i;
					break;
				}
			}
			if(j == -1){
				printf("doubt!\n");
				return 0;
			}
		}
	}
	for(i = 0; i < N; i++){
		for(j = 0; j < N; j++){
			printf("%2d", B3[i][j]);
		}
		printf("\n");
	}
*/	for(i = 0; i < N * N; i++){
		printf("%c%d\n", ansc[i], ansd[i] + 1);
	}
	return 0;
}