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 is given from Standard Input in the following format:

```
$N$
$P_2$ $P_3$ $\dots$ $P_N$
```

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

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

1
1
2
2
4
4
4
5
7
10

The following ways of choosing vertices should be counted:

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

1

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