Frog Gorf is traveling through Swamp kingdom. Unfortunately, after a poor jump, he fell into a well of $n$ meters depth. Now Gorf is on the bottom of the well and has a long way up.

The surface of the well's walls vary in quality: somewhere they are slippery, but somewhere have convenient ledges. In other words, if Gorf is on $x$ meters below ground level, then in one jump he can go up on any integer distance from $0$ to $a_x$ meters inclusive. (Note that Gorf can't jump down, only up).

Unfortunately, Gorf has to take a break after each jump (including jump on $0$ meters). And after jumping up to position $x$ meters below ground level, he'll slip exactly $b_x$ meters down while resting.

Calculate the minimum number of jumps Gorf needs to reach ground level.


-----Input-----

The first line contains a single integer $n$ ($1 \le n \le 300000$) — the depth of the well.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le i$), where $a_i$ is the maximum height Gorf can jump from $i$ meters below ground level.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($0 \le b_i \le n - i$), where $b_i$ is the distance Gorf will slip down if he takes a break on $i$ meters below ground level.


-----Output-----

If Gorf can't reach ground level, print $-1$. Otherwise, firstly print integer $k$ — the minimum possible number of jumps.

Then print the sequence $d_1,\,d_2,\,\ldots,\,d_k$ where $d_j$ is the depth Gorf'll reach after the $j$-th jump, but before he'll slip down during the break. Ground level is equal to $0$.

If there are multiple answers, print any of them.


-----Examples-----

Input
3
0 2 2
1 1 0
Output
2
1 0
Input
2
1 1
1 0
Output
-1
Input
10
0 1 2 3 5 5 6 7 8 5
9 8 7 1 5 4 3 2 0 0
Output
3
9 4 0


-----Note-----

In the first example, Gorf is on the bottom of the well and jump to the height $1$ meter below ground level. After that he slip down by meter and stays on height $2$ meters below ground level. Now, from here, he can reach ground level in one jump.

In the second example, Gorf can jump to one meter below ground level, but will slip down back to the bottom of the well. That's why he can't reach ground level.

In the third example, Gorf can reach ground level only from the height $5$ meters below the ground level. And Gorf can reach this height using a series of jumps $10 \Rightarrow 9 \dashrightarrow 9 \Rightarrow 4 \dashrightarrow 5$ where $\Rightarrow$ is the jump and $\dashrightarrow$ is slipping during breaks.

Source: codeforces | Difficulty: HARD
Tags: Graph algorithms

Graph traversal

Shortest paths

Amortized analysis

Dynamic programming

Data structures


Solutions:
from collections import deque
import os
import sys
from io import BytesIO, IOBase
BUFSIZE = 8192

class FastIO(IOBase):
	newlines = 0

	def __init__(self, file):
		self._fd = file.fileno()
		self.buffer = BytesIO()
		self.writable = 'x' in file.mode or 'r' not in file.mode
		self.write = self.buffer.write if self.writable else None

	def read(self):
		while True:
			b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
			if not b:
				break
			ptr = self.buffer.tell()
			(self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr))
		self.newlines = 0
		return self.buffer.read()

	def readline(self):
		while self.newlines == 0:
			b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
			self.newlines = b.count(b'\n') + (not b)
			ptr = self.buffer.tell()
			(self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr))
		self.newlines -= 1
		return self.buffer.readline()

	def flush(self):
		if self.writable:
			os.write(self._fd, self.buffer.getvalue())
			(self.buffer.truncate(0), self.buffer.seek(0))

class IOWrapper(IOBase):

	def __init__(self, file):
		self.buffer = FastIO(file)
		self.flush = self.buffer.flush
		self.writable = self.buffer.writable
		self.write = lambda s: self.buffer.write(s.encode('ascii'))
		self.read = lambda : self.buffer.read().decode('ascii')
		self.readline = lambda : self.buffer.readline().decode('ascii')
(sys.stdin, sys.stdout) = (IOWrapper(sys.stdin), IOWrapper(sys.stdout))
input = lambda : sys.stdin.readline().rstrip('\r\n')

class SortedList:

	def __init__(self, iterable=[], _load=200):
		values = sorted(iterable)
		self._len = _len = len(values)
		self._load = _load
		self._lists = _lists = [values[i:i + _load] for i in range(0, _len, _load)]
		self._list_lens = [len(_list) for _list in _lists]
		self._mins = [_list[0] for _list in _lists]
		self._fen_tree = []
		self._rebuild = True

	def _fen_build(self):
		self._fen_tree[:] = self._list_lens
		_fen_tree = self._fen_tree
		for i in range(len(_fen_tree)):
			if i | i + 1 < len(_fen_tree):
				_fen_tree[i | i + 1] += _fen_tree[i]
		self._rebuild = False

	def _fen_update(self, index, value):
		if not self._rebuild:
			_fen_tree = self._fen_tree
			while index < len(_fen_tree):
				_fen_tree[index] += value
				index |= index + 1

	def _fen_query(self, end):
		if self._rebuild:
			self._fen_build()
		_fen_tree = self._fen_tree
		x = 0
		while end:
			x += _fen_tree[end - 1]
			end &= end - 1
		return x

	def _fen_findkth(self, k):
		_list_lens = self._list_lens
		if k < _list_lens[0]:
			return (0, k)
		if k >= self._len - _list_lens[-1]:
			return (len(_list_lens) - 1, k + _list_lens[-1] - self._len)
		if self._rebuild:
			self._fen_build()
		_fen_tree = self._fen_tree
		idx = -1
		for d in reversed(range(len(_fen_tree).bit_length())):
			right_idx = idx + (1 << d)
			if right_idx < len(_fen_tree) and k >= _fen_tree[right_idx]:
				idx = right_idx
				k -= _fen_tree[idx]
		return (idx + 1, k)

	def _delete(self, pos, idx):
		_lists = self._lists
		_mins = self._mins
		_list_lens = self._list_lens
		self._len -= 1
		self._fen_update(pos, -1)
		del _lists[pos][idx]
		_list_lens[pos] -= 1
		if _list_lens[pos]:
			_mins[pos] = _lists[pos][0]
		else:
			del _lists[pos]
			del _list_lens[pos]
			del _mins[pos]
			self._rebuild = True

	def _loc_left(self, value):
		if not self._len:
			return (0, 0)
		_lists = self._lists
		_mins = self._mins
		(lo, pos) = (-1, len(_lists) - 1)
		while lo + 1 < pos:
			mi = lo + pos >> 1
			if value <= _mins[mi]:
				pos = mi
			else:
				lo = mi
		if pos and value <= _lists[pos - 1][-1]:
			pos -= 1
		_list = _lists[pos]
		(lo, idx) = (-1, len(_list))
		while lo + 1 < idx:
			mi = lo + idx >> 1
			if value <= _list[mi]:
				idx = mi
			else:
				lo = mi
		return (pos, idx)

	def _loc_right(self, value):
		if not self._len:
			return (0, 0)
		_lists = self._lists
		_mins = self._mins
		(pos, hi) = (0, len(_lists))
		while pos + 1 < hi:
			mi = pos + hi >> 1
			if value < _mins[mi]:
				hi = mi
			else:
				pos = mi
		_list = _lists[pos]
		(lo, idx) = (-1, len(_list))
		while lo + 1 < idx:
			mi = lo + idx >> 1
			if value < _list[mi]:
				idx = mi
			else:
				lo = mi
		return (pos, idx)

	def add(self, value):
		_load = self._load
		_lists = self._lists
		_mins = self._mins
		_list_lens = self._list_lens
		self._len += 1
		if _lists:
			(pos, idx) = self._loc_right(value)
			self._fen_update(pos, 1)
			_list = _lists[pos]
			_list.insert(idx, value)
			_list_lens[pos] += 1
			_mins[pos] = _list[0]
			if _load + _load < len(_list):
				_lists.insert(pos + 1, _list[_load:])
				_list_lens.insert(pos + 1, len(_list) - _load)
				_mins.insert(pos + 1, _list[_load])
				_list_lens[pos] = _load
				del _list[_load:]
				self._rebuild = True
		else:
			_lists.append([value])
			_mins.append(value)
			_list_lens.append(1)
			self._rebuild = True

	def discard(self, value):
		_lists = self._lists
		if _lists:
			(pos, idx) = self._loc_right(value)
			if idx and _lists[pos][idx - 1] == value:
				self._delete(pos, idx - 1)

	def remove(self, value):
		_len = self._len
		self.discard(value)
		if _len == self._len:
			raise ValueError('{0!r} not in list'.format(value))

	def pop(self, index=-1):
		(pos, idx) = self._fen_findkth(self._len + index if index < 0 else index)
		value = self._lists[pos][idx]
		self._delete(pos, idx)
		return value

	def bisect_left(self, value):
		(pos, idx) = self._loc_left(value)
		return self._fen_query(pos) + idx

	def bisect_right(self, value):
		(pos, idx) = self._loc_right(value)
		return self._fen_query(pos) + idx

	def count(self, value):
		return self.bisect_right(value) - self.bisect_left(value)

	def __len__(self):
		return self._len

	def __getitem__(self, index):
		(pos, idx) = self._fen_findkth(self._len + index if index < 0 else index)
		return self._lists[pos][idx]

	def __delitem__(self, index):
		(pos, idx) = self._fen_findkth(self._len + index if index < 0 else index)
		self._delete(pos, idx)

	def __contains__(self, value):
		_lists = self._lists
		if _lists:
			(pos, idx) = self._loc_left(value)
			return idx < len(_lists[pos]) and _lists[pos][idx] == value
		return False

	def __iter__(self):
		return (value for _list in self._lists for value in _list)

	def __reversed__(self):
		return (value for _list in reversed(self._lists) for value in reversed(_list))

	def __repr__(self):
		return 'SortedList({0})'.format(list(self))
inf = float('inf')
n = int(input())
a = [0] + list(map(int, input().split()))
b = [0] + list(map(int, input().split()))
dist = [inf] * (n + 1)
pred = [-1] * (n + 1)
dist[n] = 0
A = SortedList(list(range(0, n)))
q = deque([n])
while q:
	u0 = q.popleft()
	u = u0 + b[u0]
	idx = A.bisect_left(u - a[u])
	while idx < len(A):
		v = A[idx]
		if v <= u:
			dist[v] = 1 + dist[u0]
			A.remove(v)
			pred[v] = u0
			q.append(v)
		else:
			break
if dist[0] == inf:
	print(-1)
	exit()
ans = []
cur = 0
while not (ans and ans[-1] == cur):
	ans.append(cur)
	cur = pred[cur]
ans.pop()
ans.pop()
print(dist[0])
print(*ans[::-1])


import sys
input = sys.stdin.readline

def solve():
	n = int(input())
	A = [0] + list(map(int, input().split()))
	B = [0] + list(map(int, input().split()))
	ans = []
	(lo, hi) = (n, n - A[n])
	if hi <= 0:
		print(1)
		print(0)
		return
	while lo > 0:
		nextStep = hi
		idx = hi
		for i in range(hi, lo + 1):
			j = i + B[i]
			if nextStep > j - A[j]:
				nextStep = j - A[j]
				idx = i
		if nextStep == hi > 0:
			print(-1)
			return
		ans.append(idx)
		if nextStep <= 0:
			ans.append(0)
			print(len(ans))
			print(*ans)
			return
		(lo, hi) = (hi - 1, nextStep)
	return
solve()


from collections import deque

def solve():
	n = int(input())
	arr = [0] + list(map(int, input().split()))
	brr = [0] + list(map(int, input().split()))
	p = [-1] * (n + 1)
	p[n] = n
	bd = n
	queue = deque()
	queue.append((n, n))
	while queue:
		(a, b) = queue.popleft()
		for x in range(b - arr[b], min(b + 1, bd)):
			if p[x] == -1:
				p[x] = a
				queue.append((x, x + brr[x]))
		bd = min(bd, b - arr[b])
	if p[0] == -1:
		print(-1)
	else:
		ans = [0]
		while ans[-1] != n:
			ans.append(p[ans[-1]])
		ans.pop()
		print(len(ans))
		print(*ans[::-1])
import sys
input = lambda : sys.stdin.readline().rstrip()
solve()


import sys
input = lambda : sys.stdin.readline().rstrip()
write = lambda x: sys.stdout.write(x + '\n')
debug = lambda x: sys.stderr.write(x + '\n')
writef = lambda x: print('{:.12f}'.format(x))

class SG:

	def __init__(self, n, v=None):
		self._n = n
		self.geta = 0
		x = 0
		while 1 << x < n:
			x += 1
		self._log = x
		self._size = 1 << self._log
		self._d = [ninf] * (2 * self._size)
		if v is not None:
			for i in range(self._n):
				self._d[self._size + i] = v[i]
		for i in range(self._size - 1, 0, -1):
			self._update(i)

	def _update(self, k):
		self._d[k] = op(self._d[2 * k], self._d[2 * k + 1])

	def update(self, p, x):
		assert 0 <= p < self._n
		p += self._size
		self._d[p] = x
		for i in range(1, self._log + 1):
			k = p >> i
			self._d[k] = op(self._d[2 * k], self._d[2 * k + 1])

	def get(self, p):
		assert 0 <= p < self._n
		return self._d[p + self._size]

	def check(self):
		return [self.get(p) for p in range(self._n)]

	def query(self, left, right):
		assert 0 <= left <= right <= self._n
		sml = ninf
		smr = ninf
		left += self._size
		right += self._size
		while left < right:
			if left & 1:
				sml = op(sml, self._d[left])
				left += 1
			if right & 1:
				right -= 1
				smr = op(self._d[right], smr)
			left >>= 1
			right >>= 1
		return op(sml, smr)

	def query_all(self):
		return self._d[1]

	def max_right(self, left, f):
		if left == self._n:
			return self._n
		left += self._size
		sm = ninf
		first = True
		while first or left & -left != left:
			first = False
			while left % 2 == 0:
				left >>= 1
			if not f(op(sm, self._d[left])):
				while left < self._size:
					left *= 2
					if f(op(sm, self._d[left])):
						sm = op(sm, self._d[left])
						left += 1
				return left - self._size
			sm = op(sm, self._d[left])
			left += 1
		return self._n

	def min_left(self, right, f):
		if right == 0:
			return 0
		right += self._size
		sm = ninf
		first = True
		while first or right & -right != right:
			first = False
			right -= 1
			while right > 1 and right % 2:
				right >>= 1
			if not f(op(self._d[right], sm)):
				while right < self._size:
					right = 2 * right + 1
					if f(op(self._d[right], sm)):
						sm = op(self._d[right], sm)
						right -= 1
				return right + 1 - self._size
			sm = op(self._d[right], sm)
		return 0
op = min
ninf = 10 ** 12
n = int(input())
a = list(map(int, input().split()))
b = list(map(int, input().split()))
ns = [[] for _ in range(n + 1)]
for i in range(1, n + 1):
	ns[i + b[i - 1]].append(i)
ns[0].append(0)
sg = SG(n + 1, [0] + [i + 1 - a[i] for i in range(n)])
vs = [ninf] * (n + 1)
prv = [-1] * (n + 1)
prv[0] = 0
vs[0] = 0
from collections import deque
q = deque([0])
while q:
	u = q.popleft()
	val = vs[u]
	for v in ns[u]:
		c = v
		while c <= n:
			c = sg.max_right(c, lambda vv: vv > v)
			if c <= n:
				sg.update(c, ninf)
				vs[c] = val + 1
				prv[c] = v
				q.append(c)
if vs[n] >= ninf / 2:
	print(-1)
else:
	c = n
	ans = []
	while c != 0:
		c = prv[c]
		ans.append(c)
		c += b[c - 1]
	print(len(ans))
	write(' '.join(map(str, ans)))


def task5(n, a, b):
	F = [None] * (n + 1)
	paths = [[] for i in range(n + 1)]
	F[n] = 0
	visited = set()
	current = [n]
	high = n
	while True:
		new_vertexes = []
		for i in current:
			for j in range(i - a[i], high + 1):
				k = j + b[j]
				if k not in visited:
					new_vertexes.append(k)
					visited.add(k)
				if F[k] is None:
					paths[k] = (i, j)
					F[k] = F[i] + 1
				elif F[i] + 1 < F[k]:
					paths[k] = (i, j)
					F[k] = F[i] + 1
			high = min(high, i - a[i])
		current = new_vertexes
		if not current:
			break
	path = []
	if F[0] is not None:
		current = paths[0]
		while current[0] != n:
			path.append(current[1])
			current = paths[current[0]]
		path.append(current[1])
	return (-1 if F[0] is None else F[0], path[::-1])

def main():
	n = int(input().strip())
	arr_a = [0] + list(map(int, input().strip().split()))
	arr_b = [0] + list(map(int, input().strip().split()))
	result = task5(n, arr_a, arr_b)
	print(result[0])
	if result[1]:
		print(*result[1], sep=' ')
main()


import math
import random

def query(l, r, n, tree):
	ans = float('inf')
	idx = None
	l += n
	r += n
	prev_ans = ans
	while l <= r:
		if l & 1:
			ans = min(tree[l][0], ans)
			if prev_ans != ans:
				idx = tree[l][1]
				prev_ans = ans
			l += 1
		if r & 1 == 0:
			ans = min(tree[r][0], ans)
			if ans != prev_ans:
				idx = tree[r][1]
				prev_ans = ans
			r -= 1
		l >>= 1
		r >>= 1
	return idx

def update(i, val, n, tree):
	i += n
	tree[i][0] = val[0]
	tree[i][1] = val[1]
	i >>= 1
	while i >= 1:
		tree[i][0] = min(tree[2 * i][0], tree[2 * i + 1][0])
		if tree[i][0] == tree[2 * i][0]:
			tree[i][1] = tree[2 * i][1]
		else:
			tree[i][1] = tree[2 * i + 1][1]
		i >>= 1

def main(arr, back):
	n = len(arr)
	tree = [[float('inf'), None] for i in range(2 * n)]
	for i in range(len(arr)):
		vals = [i + back[i] - arr[i + back[i]], i]
		update(i, vals, n, tree)
	start = n - 1
	out = []
	while start != -1:
		if start - arr[start] <= -1:
			out.append(0)
			break
		next_val = query(start - arr[start], start, n, tree)
		if next_val + back[next_val] == start:
			print(-1)
			return
		else:
			out.append(next_val + 1)
			start = next_val + back[next_val]
	print(len(out))
	print(*out)
	return
n = int(input())
arr = list(map(int, input().split()))
back = list(map(int, input().split()))
main(arr, back)