Problem Statement

We have a rooted tree with $N vertices numbered 1,2,\dots,N$.
The tree is rooted at Vertex 1, and the parent of Vertex $i \ge 2$ is Vertex P_i(<i).
For each integer $k=1,2,\dots,N,$ solve the following problem:
There are 2^{k-1} ways to choose some of the vertices numbered between 1 and k so that Vertex 1 is chosen.
How many of them satisfy the following condition: the subgraph induced by the set of chosen vertices forms a perfect binary tree (with 2^d-1 vertices for a positive integer d) rooted at Vertex 1?
Since the count may be enormous, print the count modulo 998244353.

What is an induced subgraph?
Let S be a subset of the vertex set of a graph G. The subgraph H induced by this vertex set S is constructed as follows:

• Let the vertex set of H equal S.

• Then, we add edges to H as follows:

• For all vertex pairs (i, j) such that $i,j \in$ S, i < j, if there is an edge connecting i and j in G, then add an edge connecting i and j to H.

What is a perfect binary tree?

A perfect binary tree is a rooted tree that satisfies all of the following conditions:

• Every vertex that is not a leaf has exactly 2 children.
• All leaves have the same distance from the root.

Here, we regard a graph with 1 vertex and 0 edges as a perfect binary tree, too.

Input

Input is given from Standard Input in the following format:
N
$P_2 P_3 \dots P_N$

Output

Print N lines.
The $i-th (1 \le i \le N)$ line should contain the answer as an integer when k=i.

Constraints

• All values in input are integers.
• $1 \le N \le 3 \times 10^5$
• $1 \le P_i < i$

Sample Input 1

10
1 1 2 1 2 5 5 5 1

Sample Output 1

1
1
2
2
4
4
4
5
7
10

The following ways of choosing vertices should be counted:

• {1} when $k \ge 1$
• {1,2,3} when $k \ge 3$
• {1,2,5},{1,3,5} when $k \ge 5$
• {1,2,4,5,6,7,8} when $k \ge 8$
• {1,2,4,5,6,7,9},{1,2,4,5,6,8,9} when $k \ge 9$
• {1,2,10},{1,3,10},{1,5,10} when k = 10

Sample Input 2

1

Sample Output 2

1

If N=1, the 2-nd line of the Input is empty.

Sample Input 3

10
1 2 3 4 5 6 7 8 9

Sample Output 3

1
1
1
1
1
1
1
1
1
1

Sample Input 4

13
1 1 1 2 2 2 3 3 3 4 4 4

Sample Output 4

1
1
2
4
4
4
4
4
7
13
13
19
31

见题面 Input 部分

见题面 Output 部分