The zombies are gathering in their secret lair! Heidi will strike hard to destroy them once and for all. But there is a little problem... Before she can strike, she needs to know where the lair is. And the intel she has is not very good.

Heidi knows that the lair can be represented as a rectangle on a lattice, with sides parallel to the axes. Each vertex of the polygon occupies an integer point on the lattice. For each cell of the lattice, Heidi can check the level of Zombie Contamination. This level is an integer between 0 and 4, equal to the number of corners of the cell that are inside or on the border of the rectangle.

As a test, Heidi wants to check that her Zombie Contamination level checker works. Given the output of the checker, Heidi wants to know whether it could have been produced by a single non-zero area rectangular-shaped lair (with axis-parallel sides). [Image]


-----Input-----

The first line of each test case contains one integer N, the size of the lattice grid (5 ≤ N ≤ 50). The next N lines each contain N characters, describing the level of Zombie Contamination of each cell in the lattice. Every character of every line is a digit between 0 and 4.

Cells are given in the same order as they are shown in the picture above: rows go in the decreasing value of y coordinate, and in one row cells go in the order of increasing x coordinate. This means that the first row corresponds to cells with coordinates (1, N), ..., (N, N) and the last row corresponds to cells with coordinates (1, 1), ..., (N, 1).


-----Output-----

The first line of the output should contain Yes if there exists a single non-zero area rectangular lair with corners on the grid for which checking the levels of Zombie Contamination gives the results given in the input, and No otherwise.


-----Example-----
Input
6
000000
000000
012100
024200
012100
000000

Output
Yes



-----Note-----

The lair, if it exists, has to be rectangular (that is, have corners at some grid points with coordinates (x_1, y_1), (x_1, y_2), (x_2, y_1), (x_2, y_2)), has a non-zero area and be contained inside of the grid (that is, 0 ≤ x_1 < x_2 ≤ N, 0 ≤ y_1 < y_2 ≤ N), and result in the levels of Zombie Contamination as reported in the input.

Difficulty: MEDIUM_HARD

Solutions:
n = int(input())
A = [input() for i in range(n)]

def early_exit():
	print('No')
	exit()
if n < 3:
	early_exit()
corner_row = []
for i in range(n):
	if '1' in A[i]:
		corner_row.append(i)
if len(corner_row) != 2:
	early_exit()

def check_zero(x):
	for i in x:
		if i != '0':
			return False
	return True
for i in range(corner_row[0]):
	if not check_zero(A[i]):
		early_exit()
for i in range(corner_row[1] + 1, n):
	if not check_zero(A[i]):
		early_exit()
for j in range(n):
	if A[corner_row[0]][j] != '0':
		left_corner = j
		break
for j in range(n - 1, -1, -1):
	if A[corner_row[0]][j] != '0':
		right_corner = j
		break
if A[corner_row[0]][left_corner] != '1' or A[corner_row[0]][right_corner] != '1' or left_corner == right_corner:
	early_exit()
for j in range(n):
	if A[corner_row[1]][j] != '0':
		left2_corner = j
		break
for j in range(n - 1, -1, -1):
	if A[corner_row[1]][j] != '0':
		right2_corner = j
		break
if A[corner_row[1]][left2_corner] != '1' or A[corner_row[1]][right2_corner] != '1' or left_corner != left2_corner or (right_corner != right2_corner):
	early_exit()
for i in corner_row:
	for j in range(left_corner + 1, right_corner):
		if A[i][j] != '2':
			early_exit()
for i in range(corner_row[0] + 1, corner_row[1]):
	if not check_zero(A[i][:left_corner]):
		early_exit()
	if not check_zero(A[i][right_corner + 1:]):
		early_exit()
	if A[i][left_corner] != '2' or A[i][right_corner] != '2':
		early_exit()
	for j in range(left_corner + 1, right_corner):
		if A[i][j] != '4':
			early_exit()
print('Yes')


n = int(input())
aux = []
grid = []
flag = True
ans = -1
um = 0
dois = 0
quatro = 0
while n:
	n -= 1
	x = str(int(input()))
	if x != '0':
		aux.append(x)
for i in aux:
	txt = ''
	for j in i:
		if j != '0':
			txt += j
	grid.append(txt)
for i in grid:
	for j in i:
		if j == '1':
			um += 1
		if j == '2':
			dois += 1
		if j == '4':
			quatro += 1
	if ans == -1 or len(i) == ans:
		ans = len(i)
	else:
		flag = False
if um != 4 or dois != len(grid) * 2 + len(grid[0]) * 2 - 8 or quatro != len(grid) * len(grid[0]) - (len(grid) * 2 + len(grid[0]) * 2 - 4):
	flag = False
if flag:
	for i in range(0, len(grid)):
		if len(grid) - i - 1 < i:
			break
		if grid[i] != grid[len(grid) - i - 1]:
			flag = False
	for i in range(0, len(grid)):
		for j in range(0, len(grid[0])):
			if len(grid) - j - 1 < j:
				break
			if grid[i][j] != grid[i][len(grid[i]) - j - 1]:
				flag = False
if flag and ans != -1:
	print('Yes')
else:
	print('No')


import sys, math

def bfs(x, y):
	Q = [[x, y]]
	ptr = 0
	min_ = [1000, 1000]
	max_ = [0, 0]
	while ptr < len(Q):
		v = Q[ptr]
		ptr += 1
		x = v[0]
		y = v[1]
		if matrix[x][y] == 4:
			used[x][y] = 1
			if x + 1 < n:
				if used[x + 1][y] == 0:
					Q.append([x + 1, y])
					used[x + 1][y] = 1
			if x - 1 > -1:
				if used[x - 1][y] == 0:
					Q.append([x - 1, y])
					used[x - 1][y] = 1
			if y + 1 < n:
				if used[x][y + 1] == 0:
					Q.append([x, y + 1])
					used[x][y + 1] = 1
			if y - 1 > -1:
				if used[x][y - 1] == 0:
					Q.append([x, y - 1])
					used[x][y - 1] = 1
			if x < min_[0] or (x == min_[0] and y < min_[1]):
				min_ = [x, y]
			if x > max_[0] or (x == max_[0] and y > max_[1]):
				max_ = [x, y]
		else:
			used[x][y] = 0
	for i in range(min_[0], max_[0] + 1):
		for j in range(min_[1], max_[1] + 1):
			if matrix[i][j] != 4:
				print('No')
				return
	for i in range(n):
		for j in range(n):
			if used[i][j] == 0 and matrix[i][j] >= 3:
				print('No')
				return
	if min_[0] > 0:
		w = min_[0] - 1
		for j in range(min_[1], max_[1] + 1):
			used[w][j] = 1
			if matrix[w][j] != 2:
				print('No')
				return
	if max_[0] < n - 1:
		w = max_[0] + 1
		for j in range(min_[1], max_[1] + 1):
			used[w][j] = 1
			if matrix[w][j] != 2:
				print('No')
				return
	if min_[1] > 0:
		w = min_[1] - 1
		for j in range(min_[0], max_[0] + 1):
			used[j][w] = 1
			if matrix[j][w] != 2:
				print('No')
				return
	if max_[1] < n - 1:
		w = max_[1] + 1
		for j in range(min_[0], max_[0] + 1):
			used[j][w] = 1
			if matrix[j][w] != 2:
				print('No')
				return
	if min_[0] > 0 and min_[1] > 0:
		x = min_[0] - 1
		y = min_[1] - 1
		if matrix[x][y] != 1:
			print('No')
			return
		used[x][y] = 1
	if max_[0] < n - 1 and max_[1] < n - 1:
		x = max_[0] + 1
		y = max_[1] + 1
		if matrix[x][y] != 1:
			print('No')
			return
		used[x][y] = 1
	if min_[0] > 0 and max_[1] < n - 1:
		x = min_[0] - 1
		y = max_[1] + 1
		if matrix[x][y] != 1:
			print('No')
			return
		used[x][y] = 1
	if max_[0] < n - 1 and min_[1] > 0:
		x = max_[0] + 1
		y = min_[1] - 1
		if matrix[x][y] != 1:
			print('No')
			return
		used[x][y] = 1
	for i in range(n):
		for j in range(n):
			if used[i][j] == 1:
				continue
			elif matrix[i][j] != 0:
				print('No')
				return
	print('Yes')
n = int(input())
matrix = [[0] * n for i in range(n)]
for i in range(n):
	z = input()
	for j in range(n):
		matrix[i][j] = int(z[j])
used = [[0] * n for i in range(n)]
flag = 0
for i in range(n):
	for j in range(n):
		if matrix[i][j] == 4:
			flag = 1
			x = i
			y = j
			break
	if flag:
		break
if not flag:
	print('No')
	return
bfs(x, y)


n = int(input())
s = [list(map(int, input())) for _ in range(n)]
INF = 10 ** 9
(x1, x2, y1, y2) = (INF, -INF, INF, -INF)
for i in range(n):
	for j in range(n):
		if s[i][j] != 0:
			(x1, x2, y1, y2) = (min(x1, i), max(x2, i), min(y1, j), max(y2, j))
need = [[0] * n for _ in range(n)]
for i in range(x1, x2 + 1):
	for j in range(y1, y2 + 1):
		if i == x1 or i == x2:
			if j == y1 or j == y2:
				need[i][j] = 1
			else:
				need[i][j] = 2
		elif j == y1 or j == y2:
			need[i][j] = 2
		else:
			need[i][j] = 4
print('Yes' if need == s else 'No')


def main():
	size = int(input())
	sq = [[int(c) for c in input().strip()] for _ in range(size)]
	print('Yes' if haslair(sq) else 'No')

def haslair(sq):
	corners = [(r, c) for (r, row) in enumerate(sq) for (c, count) in enumerate(row) if count == 1]
	if len(corners) != 4:
		return False
	(left, right, top, bottom) = (corners[0][1], corners[3][1], corners[0][0], corners[3][0])
	if right == left + 1 or bottom == top + 1:
		return False
	for (r, row) in enumerate(sq):
		for (c, count) in enumerate(row):
			score = 3 if top < r < bottom else 1 if r in (top, bottom) else 0
			score += 3 if left < c < right else 1 if c in (left, right) else 0
			if count != (0, 0, 1, 0, 2, None, 4)[score]:
				return False
	return True
main()


N = int(input())
grid = []
x1 = 50
y1 = 50
x2 = -1
y2 = -1
for y in range(N):
	grid.append(list(map(int, input())))
	for (x, num) in enumerate(grid[-1]):
		if num == 4:
			x1 = min(x1, x)
			y1 = min(y1, y)
			x2 = max(x2, x)
			y2 = max(y2, y)
if x1 == 51:
	print('No')
else:
	for y in range(N):
		for x in range(N):
			ex = 0
			if x1 <= x <= x2 and y1 <= y <= y2:
				ex = 4
			elif (x == x1 - 1 or x == x2 + 1) and y1 <= y <= y2:
				ex = 2
			elif (y == y1 - 1 or y == y2 + 1) and x1 <= x <= x2:
				ex = 2
			elif (x == x1 - 1 or x == x2 + 1) and (y == y1 - 1 or y == y2 + 1):
				ex = 1
			if ex != grid[y][x]:
				print('No')
				break
		else:
			continue
		break
	else:
		print('Yes')


from sys import stdin
n = int(stdin.readline())
nn = [[int(x) for x in stdin.readline().rstrip()] for i in range(n)]

def run():
	for row in range(n):
		for col in range(n):
			if nn[row][col] == 0:
				continue
			if countAdj(row, col, n - 1):
				print('No')
				return 0
	print('Yes')

def countAdj(row, col, n):
	cnt = 0
	adj = 0
	if row > 1 and nn[row - 1][col]:
		cnt += 1
		adj += nn[row - 1][col]
	if row < n and nn[row + 1][col]:
		cnt += 1
		adj += nn[row + 1][col]
	if col > 1 and nn[row][col - 1]:
		cnt += 1
		adj += nn[row][col - 1]
	if col < n and nn[row][col + 1]:
		cnt += 1
		adj += nn[row][col + 1]
	if cnt <= 1:
		return True
	elif cnt < 4 and nn[row][col] == cnt - 1:
		if nn[row][col] == 1 and adj != 4:
			return True
		if nn[row][col] == 2 and (not 6 <= adj <= 8):
			return True
		return False
	elif cnt == 4 and nn[row][col] == cnt:
		if nn[row][col] == 4 and (not (adj == 8 or adj == 10 or adj == 12 or (adj == 14) or (adj == 16))):
			return True
		return False
	return True
run()


def check_cell(i, j, n):
	k = 0
	if i > 1 and field[i - 1][j]:
		k += 1
	if i < n - 1 and field[i + 1][j]:
		k += 1
	if j > 1 and field[i][j - 1]:
		k += 1
	if j < n - 1 and field[i][j + 1]:
		k += 1
	if k < 2:
		return 0
	elif k < 4:
		return k - 1
	else:
		return 4
n = int(input())
field = []
for i in range(n):
	temp = [int(x) for x in input()]
	field.append(temp)

def main():
	for i in range(n):
		for j in range(n):
			if field[i][j] and (not field[i][j] == check_cell(i, j, n)):
				print('No')
				return 0
	print('Yes')
main()