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.

Constraints

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

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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 $k=i$.

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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 $k = 10$

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Sample Input 2

1

Sample Output 2

1

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

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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

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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