The country has n cities and n - 1 bidirectional roads, it is possible to get from every city to any other one if you move only along the roads. The cities are numbered with integers from 1 to n inclusive.

All the roads are initially bad, but the government wants to improve the state of some roads. We will assume that the citizens are happy about road improvement if the path from the capital located in city x to any other city contains at most one bad road.

Your task is — for every possible x determine the number of ways of improving the quality of some roads in order to meet the citizens' condition. As those values can be rather large, you need to print each value modulo 1 000 000 007 (10^9 + 7).


-----Input-----

The first line of the input contains a single integer n (2 ≤ n ≤ 2·10^5) — the number of cities in the country. Next line contains n - 1 positive integers p_2, p_3, p_4, ..., p_{n} (1 ≤ p_{i} ≤ i - 1) — the description of the roads in the country. Number p_{i} means that the country has a road connecting city p_{i} and city i. 


-----Output-----

Print n integers a_1, a_2, ..., a_{n}, where a_{i} is the sought number of ways to improve the quality of the roads modulo 1 000 000 007 (10^9 + 7), if the capital of the country is at city number i.


-----Examples-----
Input
3
1 1

Output
4 3 3
Input
5
1 2 3 4

Output
5 8 9 8 5

Difficulty: HARD
Tags: Tree algorithms

Dynamic programming
Skill types: Dynamic programming

Starter code:

Solutions:
class Graph:

	def __init__(self, n_vertices, edges, directed=True, weighted=False):
		self.n_vertices = n_vertices
		self.edges = edges
		self.directed = directed
		self.weighted = weighted

	@property
	def adj(self):
		try:
			return self._adj
		except AttributeError:
			adj = [[] for _ in range(self.n_vertices)]

			def d_w(e):
				adj[e[0]].append((e[1], e[2]))

			def ud_w(e):
				adj[e[0]].append((e[1], e[2]))
				adj[e[1]].append((e[0], e[2]))

			def d_uw(e):
				adj[e[0]].append(e[1])

			def ud_uw(e):
				adj[e[0]].append(e[1])
				adj[e[1]].append(e[0])
			helper = (ud_uw, d_uw, ud_w, d_w)[self.directed + self.weighted * 2]
			for e in self.edges:
				helper(e)
			self._adj = adj
			return adj

class RootedTree(Graph):

	def __init__(self, n_vertices, edges, root_vertex):
		self.root = root_vertex
		super().__init__(n_vertices, edges, False, False)

	@property
	def parent(self):
		try:
			return self._parent
		except AttributeError:
			adj = self.adj
			parent = [None] * self.n_vertices
			parent[self.root] = -1
			stack = [self.root]
			for i in range(self.n_vertices):
				v = stack.pop()
				for u in adj[v]:
					if parent[u] is None:
						parent[u] = v
						stack.append(u)
			self._parent = parent
			return parent

	@property
	def children(self):
		try:
			return self._children
		except AttributeError:
			children = [None] * self.n_vertices
			for (v, (l, p)) in enumerate(zip(self.adj, self.parent)):
				children[v] = [u for u in l if u != p]
			self._children = children
			return children

	@property
	def dfs_order(self):
		try:
			return self._dfs_order
		except AttributeError:
			order = [None] * self.n_vertices
			children = self.children
			stack = [self.root]
			for i in range(self.n_vertices):
				v = stack.pop()
				order[i] = v
				for u in children[v]:
					stack.append(u)
			self._dfs_order = order
			return order
from functools import reduce
from itertools import accumulate, chain

def rerooting(rooted_tree, merge, identity, finalize):
	N = rooted_tree.n_vertices
	parent = rooted_tree.parent
	children = rooted_tree.children
	order = rooted_tree.dfs_order
	dp_down = [None] * N
	for v in reversed(order[1:]):
		dp_down[v] = finalize(reduce(merge, (dp_down[c] for c in children[v]), identity))
	dp_up = [None] * N
	dp_up[0] = identity
	for v in order:
		if len(children[v]) == 0:
			continue
		temp = (dp_up[v],) + tuple((dp_down[u] for u in children[v])) + (identity,)
		left = tuple(accumulate(temp, merge))
		right = tuple(accumulate(reversed(temp[2:]), merge))
		for (u, l, r) in zip(children[v], left, reversed(right)):
			dp_up[u] = finalize(merge(l, r))
	res = [None] * N
	for (v, l) in enumerate(children):
		res[v] = reduce(merge, (dp_down[u] for u in children[v]), identity)
		res[v] = finalize(merge(res[v], dp_up[v]))
	return res

def solve(T):
	MOD = 10 ** 9 + 7

	def merge(x, y):
		return x * y % MOD

	def finalize(x):
		return x + 1
	return [v - 1 for v in rerooting(T, merge, 1, finalize)]

def __starting_point():
	N = int(input())
	edges = [(i + 1, p - 1) for (i, p) in enumerate(map(int, input().split()))]
	T = RootedTree(N, edges, 0)
	print(*solve(T))
__starting_point()