Score : $700$ points

Problem Statement

We have a rooted tree with $N$ vertices numbered $1$ to $N$. Vertex $1$ is the root of the tree, and the parent of vertex $i\ (2\leq i)$ is vertex $p_i$.

Each vertex has a color: black or white. Initially, all vertices are white.

In this rooted tree, vertex $i$ is said to be good when all vertices on the simple path connecting vertices $1$ and $i$ (including vertices $1$ and $i$) are black, and bad otherwise.

Until all vertices are black, let us repeat the following operation: choose one vertex from the bad vertices uniformly at random, and repaint the chosen vertex black.

Find the expected value of the number of times we perform the operation, modulo $998244353$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq p_i < i$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$

$p_2$ $p_3$ $\dots$ $p_{N}$

Output

Print the answer.

4
1 1 3

831870300

Consider a case where the first three operations have chosen vertices $1$, $2$, and $4$ in this order. Then, vertices $1$ and $2$ are good, but vertex $4$ is still bad since vertex $3$, an ancestor of vertex $4$, is white. Thus, the fourth operation will choose vertex $3$ or $4$ uniformly at random.

The expected value of the number of times we perform the operation is $\displaystyle \frac{35}{6}$.

15
1 2 1 1 4 5 3 3 5 10 3 6 3 13

515759610