In a galaxy far, far away Lesha the student has just got to know that he has an exam in two days. As always, he hasn't attended any single class during the previous year, so he decided to spend the remaining time wisely.

Lesha knows that today he can study for at most a hours, and he will have b hours to study tomorrow. Note that it is possible that on his planet there are more hours in a day than on Earth. Lesha knows that the quality of his knowledge will only depend on the number of lecture notes he will read. He has access to an infinite number of notes that are enumerated with positive integers, but he knows that he can read the first note in one hour, the second note in two hours and so on. In other words, Lesha can read the note with number k in k hours. Lesha can read the notes in arbitrary order, however, he can't start reading a note in the first day and finish its reading in the second day.

Thus, the student has to fully read several lecture notes today, spending at most a hours in total, and fully read several lecture notes tomorrow, spending at most b hours in total. What is the maximum number of notes Lesha can read in the remaining time? Which notes should he read in the first day, and which — in the second?

Input

The only line of input contains two integers a and b (0 ≤ a, b ≤ 10^{9}) — the number of hours Lesha has today and the number of hours Lesha has tomorrow.

Output

In the first line print a single integer n (0 ≤ n ≤ a) — the number of lecture notes Lesha has to read in the first day. In the second line print n distinct integers p_1, p_2, …, p_n (1 ≤ p_i ≤ a), the sum of all p_i should not exceed a.

In the third line print a single integer m (0 ≤ m ≤ b) — the number of lecture notes Lesha has to read in the second day. In the fourth line print m distinct integers q_1, q_2, …, q_m (1 ≤ q_i ≤ b), the sum of all q_i should not exceed b.

All integers p_i and q_i should be distinct. The sum n + m should be largest possible.

Examples

Input

3 3


Output

1
3 
2
2 1 

Input

9 12


Output

2
3 6
4
1 2 4 5

Note

In the first example Lesha can read the third note in 3 hours in the first day, and the first and the second notes in one and two hours correspondingly in the second day, spending 3 hours as well. Note that Lesha can make it the other way round, reading the first and the second notes in the first day and the third note in the second day.

In the second example Lesha should read the third and the sixth notes in the first day, spending 9 hours in total. In the second day Lesha should read the first, second fourth and fifth notes, spending 12 hours in total.

Difficulty: MEDIUM_HARD
Tags: Greedy algorithms
Skill types: Greedy algorithms

Starter code:

Solutions:
def get_out(list):
	print(len(list))
	out = ' '.join(list)
	print(out)
arr = input().split()
a = int(arr[0])
b = int(arr[1])
s = a + b
temp = 0
i = 0
while temp <= s:
	i += 1
	temp += i
i -= 1
list_a = []
list_b = []
for x in range(i, 0, -1):
	if a - x >= 0:
		a -= x
		list_a.append(str(x))
	elif b - x >= 0:
		b -= x
		list_b.append(str(x))
get_out(list_a)
get_out(list_b)


(a, b) = map(int, input().split())
s = a + b
best = 0
for i in range(1, s + 1):
	if i * (i + 1) // 2 <= s:
		best = i
	else:
		break
save_best = best
first_day = set()
second_day = set()
while a != 0 and best != 0:
	nxt = min(a, best)
	best -= 1
	first_day.add(nxt)
	a -= nxt
for i in range(1, save_best + 1):
	if i not in first_day:
		second_day.add(i)
print(len(first_day))
print(' '.join(map(str, sorted(first_day))))
print(len(second_day))
print(' '.join(map(str, sorted(second_day))))


import collections, atexit, math, sys, bisect
sys.setrecursionlimit(1000000)

def getIntList():
	return list(map(int, input().split()))
try:
	import numpy

	def dprint(*args, **kwargs):
		print(*args, **kwargs, file=sys.stderr)
	dprint('debug mode')
except Exception:

	def dprint(*args, **kwargs):
		pass
inId = 0
outId = 0
if inId > 0:
	dprint('use input', inId)
	sys.stdin = open('input' + str(inId) + '.txt', 'r')
if outId > 0:
	dprint('use output', outId)
	sys.stdout = open('stdout' + str(outId) + '.txt', 'w')
	atexit.register(lambda : sys.stdout.close())
(A, B) = getIntList()
tot = 0
for i in range(1, 1000000):
	tot += i
	if tot > A + B:
		break
n = i - 1
zr = [1 for i in range(n + 1)]
tot = 0
zz = []
for i in range(1, 1000000):
	tot += i
	if tot > A:
		tot -= i
		break
	zz.append(i)
if zz and zz[-1] < n:
	for i in range(A - tot):
		zz[-1 - i] += 1
for x in zz:
	zr[x] = 0
print(len(zz))
for x in zz:
	print(x, end=' ')
print()
print(n - len(zz))
for i in range(1, n + 1):
	if zr[i] == 1:
		print(i, end=' ')
print()


from math import sqrt

def cram_time():
	(a, b) = map(int, input().split())
	n = int(sqrt(2 * (a + b)) - 2)
	while (n + 1) * (n + 2) / 2 <= a + b:
		n += 1
	tasks = list(range(n, 0, -1))
	t1 = []
	t2 = []
	while tasks:
		current_task = tasks.pop(0)
		if current_task <= a:
			t1.append(current_task)
			a -= current_task
		else:
			t2.append(current_task)
			b -= current_task
	print(len(t1))
	print(' '.join(list(map(str, t1))))
	print(len(t2))
	print(' '.join(list(map(str, t2))))
cram_time()


def main():
	(a, b) = map(int, input().split())
	if a + b == 0:
		print(0)
		print(0)
		return
	_sum = 0
	last = 0
	for i in range(1, a + b + 1):
		_sum += i
		if _sum == a + b:
			last = i
			break
		elif _sum > a + b:
			_sum -= i
			last = i - 1
			break
	first = 1
	current = last
	used = set()
	q_1 = min(a, b)
	q = q_1
	if q <= last:
		if q == 0:
			pass
		else:
			used.add(q)
	else:
		while q >= current:
			used.add(current)
			q -= current
			current -= 1
		if q != 0:
			used.add(q)
	if q_1 == a:
		print(len(used))
		print(*used)
		print(last - len(used))
		for k in range(1, last + 1):
			if k not in used:
				print(k, end=' ')
		return
	else:
		z = last - len(used)
		print(last - len(used))
		for k in range(1, last + 1):
			if k not in used:
				z -= 1
				if z == 0:
					print(k)
				else:
					print(k, end=' ')
		print(len(used))
		print(*used)
		return
main()


def print_set(n):
	print(len(n))
	n = [str(i) for i in n]
	print(' '.join(n))
(a, b) = map(int, input().split())
s = a + b
used = 0
k = 1
A = []
B = []
while used + k <= s:
	used += k
	k += 1
k -= 1
while k > 0:
	if k <= a:
		A.append(k)
		a -= k
	else:
		b -= k
		B.append(k)
	k -= 1
print_set(A)
print_set(B)