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.
• $1 \le N \le 3 \times 10^5$
• $1 \le P_i < i$

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 $k=i$.