Chef is an advocate for Go Green Initiative. Today he had n trees planted in a row outside his his restaurant. Today, the height of i-th tree is hi feet. The trees grow at a rate of mi feet per day.
Chef knows that trees will look beautiful if they form a zig-zag sequence. The trees will be said to be in Zig-zag sequence if the heights of tree first increases or decreases, then alternates between decreasing/increasing respectively. Formally, the trees will be said to in Zig-zag sequence if one of the following two conditions holds.

- h1 < h2 > h3 < h4 and so on..
- h1 >  h2 < h3 > h4 and so on..

Chef wants to know intervals of time when the heights of the trees will form a zig-zag sequence.

-----Input-----
The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows.
The first line of each test case contains a single integer n, denoting the number of trees.
The ith of following N lines contains two space separated integers hi and mi, denoting the initial height and the growth speed for ith tree.

-----Output-----
For each test case, output an integer Q - the amount of the periods of consecutive moments of time, when the trees for a zig-zag sequence.
On the following Q lines, output the intervals of time when the trees' heights form a zig-zag sequence. For each intervals, output its' smallest and the largest instants of time. If the range is infinite, output Inf as the right bound.
The test cases are designed in such a way that the total output won't exceed 2 MB.

-----Constraints-----

- 1 ≤ T ≤ 105
- 1 ≤ n ≤ 10
- Subtask 1 (23 points): 0 ≤ hi, mi ≤ 10
- Subtask 2 (77 points): 0 ≤ hi, mi ≤ 109
- 1 ≤ sum of n over a test cases in a single test file ≤ 5 × 105

-----Example-----
Input:3
3
0 1
2 2
0 3
2
2 1
1 2
3
1 1
2 2
3 3

Output:1
0 1
2
0 0
2 Inf
0

-----Explanation-----
Example case 1. In the first case 0 2 0 is already a zig-zag sequence, but on the 2nd second it will become 2 6 6 and will never turn back into zig-zag

Difficulty: UNKNOWN_DIFFICULTY

Starter code:

Solutions:
def get(l, n):
	(l1, l2) = ([], [])
	i = 1
	(h1, m1) = l[0]
	while i < len(l):
		(h2, m2) = l[i]
		if h1 > h2:
			if m1 >= m2:
				l1 += [(0, 10 ** 20)]
				l2 += [(-1, -1)]
			else:
				d = (h1 - h2) // (m2 - m1)
				if (h1 - h2) % (m2 - m1) == 0:
					l1 += [(0, d - 1)]
					l2 += [(d + 1, 10 ** 20)]
				else:
					l1 += [(0, d)]
					l2 += [(d + 1, 10 ** 20)]
		elif h1 == h2:
			if m1 > m2:
				l1 += [(1, 10 ** 20)]
				l2 += [(-1, -1)]
			elif m1 == m2:
				l1 += [(-1, -1)]
				l2 += [(-1, -1)]
			else:
				l2 += [(1, 10 ** 20)]
				l1 += [(-1, -1)]
		elif m1 <= m2:
			l2 += [(0, 10 ** 20)]
			l1 += [(-1, -1)]
		else:
			d = (h2 - h1) // (m1 - m2)
			if (h2 - h1) % (m1 - m2) == 0:
				l2 += [(0, d - 1)]
				l1 += [(d + 1, 10 ** 20)]
			else:
				l2 += [(0, d)]
				l1 += [(d + 1, 10 ** 20)]
		i += 1
		(h1, m1) = (h2, m2)
	return (l1, l2)

def intersect(k1, k2):
	(k1, k2) = (min(k1, k2), max(k1, k2))
	(c1, c2) = k1
	(c3, c4) = k2
	l = [c1, c2, c3, c4]
	l.sort()
	if l[2] == c2:
		return (c3, min(c2, c4))
	elif l[3] == c2:
		return k2
	else:
		return (-1, -1)

def union(k1, k2):
	(k1, k2) = (min(k1, k2), max(k1, k2))
	(c1, c2) = k1
	(c3, c4) = k2
	l = [c1, c2, c3, c4]
	l.sort()
	if c2 == l[3]:
		return [c1, c2]
	elif c2 == l[2] or c3 - c2 == 1:
		return [c1, c4]
	else:
		return [c1, c2, c3, c4]

def aa(l1, l2, n):
	(c1, c2) = (0, 10 ** 20)
	i = 0
	n -= 1
	while i < n:
		if i % 2 == 0:
			(k1, k2) = l1[i]
		else:
			(k1, k2) = l2[i]
		i += 1
		if (k1, k2) == (-1, -1):
			return (-1, -1)
		(c1, c2) = intersect((c1, c2), (k1, k2))
		if (c1, c2) == (-1, -1):
			return (c1, c2)
	return (c1, c2)
test = int(input())
while test != 0:
	test -= 1
	n = int(input())
	l = []
	i = 0
	while i < n:
		(c1, c2) = list(map(int, input().split()))
		l += [(c1, c2)]
		i += 1
	if n == 1:
		print(1)
		print('0 Inf')
	else:
		(l1, l2) = get(l, n)
		(k1, k2) = aa(l1, l2, n)
		if (k1, k2) == (-1, -1):
			(k1, k2) = aa(l2, l1, n)
			if (k1, k2) == (-1, -1):
				print(0)
			else:
				print(1)
				if k2 == 10 ** 20:
					k2 = 'Inf'
				print(str(k1) + ' ' + str(k2))
		else:
			(k3, k4) = aa(l2, l1, n)
			if (k3, k4) == (-1, -1):
				print(1)
				if k2 == 10 ** 20:
					k2 = 'Inf'
				print(str(k1) + ' ' + str(k2))
			else:
				p = union((k1, k2), (k3, k4))
				if len(p) == 2:
					(c1, c2) = p
					if c2 == 10 ** 20:
						c2 = 'Inf'
					print(1)
					print(str(c1) + ' ' + str(c2))
				else:
					(c1, c2, c3, c4) = p
					if c4 == 10 ** 20:
						c4 = 'Inf'
					print(2)
					print(str(c1) + ' ' + str(c2))
					print(str(c3) + ' ' + str(c4))


import math

def CHAHG1(xh, m):
	x = [x for x in xh]
	y = [y for y in m]
	up = []
	lw = []
	L = None
	R = None
	for i in range(0, len(x) - 1):
		if i % 2 == 0:
			if y[i] - y[i + 1] == 0:
				if x[i] < x[i + 1]:
					lw.append(0)
				else:
					return False
			elif y[i] < y[i + 1]:
				l = float(x[i + 1] - x[i]) / (y[i] - y[i + 1])
				l = int(math.floor(l)) + 1
				lw.append(max(0, l))
			else:
				r = float(x[i + 1] - x[i]) / (y[i] - y[i + 1])
				r = int(math.ceil(r)) - 1
				if r < 0:
					return False
				up.append(r)
		elif y[i] - y[i + 1] == 0:
			if x[i] > x[i + 1]:
				lw.append(0)
			else:
				return False
		elif y[i] > y[i + 1]:
			l = float(x[i + 1] - x[i]) / (y[i] - y[i + 1])
			l = int(math.floor(l)) + 1
			lw.append(max(0, l))
		else:
			r = float(x[i + 1] - x[i]) / (y[i] - y[i + 1])
			r = int(math.ceil(r)) - 1
			if r < 0:
				return False
			up.append(r)
	if len(lw) > 0:
		L = max(lw)
	else:
		L = 0
	if len(up) > 0:
		R = min(up)
		if L > R:
			return False
	else:
		R = 'Inf'
	return (L, R)

def CHAHG2(xh, m):
	x = [x for x in xh]
	y = [y for y in m]
	up = []
	lw = []
	L = None
	R = None
	for i in range(0, len(x) - 1):
		if i % 2 == 1:
			if y[i] - y[i + 1] == 0:
				if x[i] < x[i + 1]:
					lw.append(0)
				else:
					return False
			elif y[i] < y[i + 1]:
				l = float(x[i + 1] - x[i]) / (y[i] - y[i + 1])
				l = int(math.floor(l)) + 1
				lw.append(max(0, l))
			else:
				r = float(x[i + 1] - x[i]) / (y[i] - y[i + 1])
				r = int(math.ceil(r)) - 1
				if r < 0:
					return False
				up.append(r)
		elif y[i] - y[i + 1] == 0:
			if x[i] > x[i + 1]:
				lw.append(0)
			else:
				return False
		elif y[i] > y[i + 1]:
			l = float(x[i + 1] - x[i]) / (y[i] - y[i + 1])
			l = int(math.floor(l)) + 1
			lw.append(max(0, l))
		else:
			r = float(x[i + 1] - x[i]) / (y[i] - y[i + 1])
			r = int(math.ceil(r)) - 1
			if r < 0:
				return False
			up.append(r)
	if len(lw) > 0:
		L = max(lw)
	else:
		L = 0
	if len(up) > 0:
		R = min(up)
		if L > R:
			return False
	else:
		R = 'Inf'
	return (L, R)
T = int(input())
while T:
	T = T - 1
	N = int(input())
	xh = []
	m = []
	for i in range(N):
		(a, b) = list(map(int, input().split()))
		xh.append(a)
		m.append(b)
	res = []
	p = CHAHG1(xh, m)
	q = CHAHG2(xh, m)
	if p is not False:
		res.append(p)
	if q is not False:
		res.append(q)
	res.sort()
	sz = len(res)
	if N == 1:
		print('1')
		print('0 Inf')
	elif sz == 2 and res[0][1] + 1 == res[1][0]:
		print('1')
		print(res[0][0], res[1][1])
	else:
		print(sz)
		for (L, R) in res:
			print(L, R)


from random import *

def check(h):
	ans = True
	for i in range(n - 1):
		if i % 2 == 0:
			if h[i] < h[i + 1]:
				ans = True
			else:
				ans = False
				break
		elif h[i] > h[i + 1]:
			ans = True
		else:
			ans = False
			break
	if ans == True:
		return 1
	ans = True
	for i in range(n - 1):
		if i % 2 == 1:
			if h[i] < h[i + 1]:
				ans = True
			else:
				ans = False
				break
		elif h[i] > h[i + 1]:
			ans = True
		else:
			ans = False
			break
	if ans == True:
		return 1
	return 0
for t in range(int(input())):
	n = int(input())
	h = []
	r = []
	for i in range(n):
		(x, y) = list(map(int, input().split()))
		h.append(x)
		r.append(y)
	ans = []
	curr = check(h)
	if curr != 0:
		ans.append(0)
	for i in range(1, 20):
		for j in range(n):
			h[j] += r[j]
		temp = check(h)
		if temp != curr:
			if temp == 0:
				ans.append(i - 1)
			elif temp == 1 and curr == 0:
				ans.append(i)
			elif temp == 2 and curr == 0:
				ans.append(i)
			elif temp == 1 and curr == 2:
				ans.append(i - 1)
				ans.append(i)
			elif temp == 2 and curr == 1:
				ans.append(i - 1)
				ans.append(i)
		curr = temp
		if len(ans) == 3:
			break
	if len(ans) % 2 == 1:
		ans.append('Inf')
	print(len(ans) / 2)
	for i in range(0, len(ans), 2):
		print(ans[i], ans[i + 1])


def is_zigzag(h):
	if len(h) <= 1:
		return True
	sign = h[1] - h[0]
	if len(h) == 2 and sign == 0:
		return False
	for i in range(2, len(h)):
		diff = h[i] - h[i - 1]
		if diff * sign >= 0:
			return False
		sign = diff
	return True

def is_possible(h, r):
	for i in range(1, len(h)):
		diff = h[i] - h[i - 1]
		if diff < 0 and r[i] > r[i - 1]:
			return True
		elif diff > 0 and r[i] < r[i - 1]:
			return True
		elif diff == 0 and r[i] != r[i - 1]:
			return True
	return False

def add_day(h, r):
	for i in range(len(h)):
		h[i] += r[i]
INF = float('inf')

def find_intervals(heights, rates):
	intervals = []
	f = is_zigzag(heights)
	if f:
		interval = [0, 'Inf']
		intervals.append(interval)
	day = 0
	while is_possible(heights, rates):
		day += 1
		add_day(heights, rates)
		if is_zigzag(heights):
			if f == 1:
				continue
			else:
				intervals.append([day, 'Inf'])
				f = 1
		elif f == 1:
			intervals[-1][-1] = day - 1
			f = 0
	return intervals
t = int(input())
for i in range(t):
	n = int(input())
	heights = []
	rates = []
	for k in range(n):
		(h, m) = list(map(int, input().split()))
		heights.append(h)
		rates.append(m)
	intervals = find_intervals(heights, rates)
	print(len(intervals))
	for interval in intervals:
		print('{} {}'.format(interval[0], interval[1]))


from math import ceil
t = int(input())
rangex1 = int(0)
rangex2 = int(0)
rangey1 = int(0)
rangey2 = int(0)
while t > 0:
	n = int(input())
	h = [int(0)] * 11
	m = [int(0)] * 11
	alt = int(-1)
	for i in range(0, n):
		(a, b) = input().split(' ')
		h[i] = int(a)
		m[i] = int(b)
	if n == 1:
		print('1')
		print('0 Inf')
	else:
		state = int(1)
		count = int(0)
		index = int(0)
		curtime = int(0)
		while True:
			state = 1
			alt = -1
			for i in range(1, n):
				if i == 1:
					if h[i] > h[i - 1]:
						alt = 1
					elif h[i] < h[i - 1]:
						alt = 0
					else:
						alt = 2
						state = 0
						break
				elif alt == 1:
					if h[i] < h[i - 1]:
						alt = 0
					else:
						state = 0
						break
				elif h[i] > h[i - 1]:
					alt = 1
				else:
					state = 0
					break
			if state == 1:
				count += 1
				if index == 0:
					rangex1 = int(curtime)
				else:
					rangex2 = int(curtime)
			mintime = int(0)
			found = int(0)
			tt = int(0)
			for i in range(1, n):
				if h[i - 1] < h[i]:
					if m[i - 1] > m[i]:
						tt = int(ceil(float(h[i] - h[i - 1]) / float(m[i - 1] - m[i])))
						if found == 0:
							mintime = tt
							found = 1
						elif tt < mintime:
							mintime = tt
				elif h[i - 1] > h[i]:
					if m[i] > m[i - 1]:
						tt = int(ceil(float(h[i - 1] - h[i]) / float(m[i] - m[i - 1])))
						if found == 0:
							mintime = tt
							found = 1
						elif tt < mintime:
							mintime = tt
				elif m[i] != m[i - 1]:
					tt = 1
					if found == 0:
						mintime = tt
						found = 1
					elif tt < mintime:
						mintime = tt
			if found == 0:
				if state == 1:
					if index == 0:
						rangey1 = int(-1)
						index += 1
					else:
						rangey2 = int(-1)
						index += 1
				break
			else:
				if state == 1:
					if index == 0:
						rangey1 = int(curtime + mintime - 1)
						index += 1
					else:
						rangey2 = int(curtime + mintime - 1)
						index += 1
				curtime += mintime
				for i in range(0, n):
					h[i] += m[i] * mintime
		if count > 1:
			if rangey1 == rangex2 - 1:
				rangey1 = rangey2
				count -= 1
		print(count)
		if count >= 1:
			if rangey1 == -1:
				print('%d Inf' % rangex1)
			else:
				print(rangex1, rangey1)
		if count >= 2:
			if rangey2 == -1:
				print('%d Inf' % rangex2)
			else:
				print(rangey1, rangey2)
	t -= 1


import math

def floorm(a, b):
	if a % b == 0:
		return a / b - 1
	else:
		return a / b

def ceilp(a, b):
	if a % b == 0:
		return a / b + 1
	else:
		return (a + 1) / b + 1

def calc(h, m, f):
	(l, r) = (0, 1234567890)
	for i in range(len(h) - 1):
		if i & 1:
			(u, v) = (i, i + 1)
		else:
			(u, v) = (i + 1, i)
		if f:
			(u, v) = (v, u)
		(H, M) = (h[u] - h[v], m[v] - m[u])
		if M > 0:
			r = min(floorm(H, M), r)
		elif M < 0:
			l = max(ceilp(H, M), l)
		elif H <= 0:
			return (-1, -1)
		if l > r:
			return (-1, -1)
	return (l, r)

def printerval(l, r):
	if r == 1234567890:
		r = 'Inf'
	print(l, r)

def doit():
	(n, h, m, a, t, al, ar) = (int(input()), [], [], [], [0], [], [])
	for i in range(n):
		h += [0]
		m += [0]
		a += [0]
	for i in range(n):
		(h[i], m[i]) = list(map(int, input().split()))
	if n == 1:
		print(1)
		print('0 Inf')
		return
	(l1, r1) = calc(h, m, 1)
	(l2, r2) = calc(h, m, 0)
	if l1 != -1 and l2 != -1:
		if l1 > l2:
			(l1, l2, r1, r2) = (l2, l1, r2, r1)
		if r1 + 1 == l2:
			print(1)
			printerval(l1, r2)
		else:
			print(2)
			printerval(l1, r1)
			printerval(l2, r2)
	elif l1 != -1:
		print(1)
		printerval(l1, r1)
	elif l2 != -1:
		print(1)
		printerval(l2, r2)
	else:
		print(0)
for cas in range(int(input())):
	doit()


def is_zig_zag(h):
	if len(h) < 2:
		return True
	if h[0] == h[1]:
		return False
	fs = (lambda x, y: x < y, lambda x, y: x > y)
	f = 0 if h[0] < h[1] else 1
	for i in range(1, len(h) - 1):
		if not fs[(i + f) % 2](h[i], h[i + 1]):
			return False
	return True

def t_diff(h, m):
	max_t_diff = None
	for i in range(len(h) - 1):
		(h_diff, m_diff) = (h[i] - h[i + 1], m[i] - m[i + 1])
		if h_diff == m_diff or m_diff == 0:
			continue
		if h_diff == 0:
			return 1
		if (h_diff < 0) != (m_diff < 0):
			local_t_diff = max(1, abs(h_diff / m_diff))
			if max_t_diff is None:
				max_t_diff = local_t_diff
			else:
				max_t_diff = min(max_t_diff, local_t_diff)
	return max_t_diff
for _ in range(eval(input())):
	n = eval(input())
	(h, m) = list(map(list, list(zip(*[list(map(int, input().split())) for _ in range(n)]))))
	t = 0
	zig_zags = []
	zig_zag_start = None
	while True:
		if is_zig_zag(h):
			if zig_zag_start is None:
				zig_zag_start = t
		elif zig_zag_start is not None:
			zig_zags.append((zig_zag_start, t - 1))
			zig_zag_start = None
		td = t_diff(h, m)
		if td is None:
			if zig_zag_start is not None:
				zig_zags.append((zig_zag_start, 'Inf'))
				zig_zag_start = None
			break
		h = list(map(lambda x, y: x + td * y, h, m))
		t += td
	print(len(zig_zags))
	for (s, e) in zig_zags:
		print(s, e)