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Problem Statement

For a positive integer $N$, a rooted binary tree that satisfies the following conditions is said to be a beautiful binary tree of degree $N$.

• Each vertex has $0$ or $1$ written on it.
• Each vertex that is a leaf has $1$ written on it.
• It is possible to do the following operation at most $N-1$ times so that the root has $N$ written on it and the other vertices have $0$ written on them.
  • Choose vertices $u$ and $v$, where $v$ must be a child of $u$ or a child of "a child of $u$." Let $a_u \gets a_u + a_v, a_v \gets 0$, where $a_u$ and $a_v$ are the numbers written on $u$ and $v$, respectively.

Given $N$, find the number, modulo $998244353$, of beautiful binary trees of degree $N$.

Constraints

• $1 \leq N \leq 10^7$
• All values in input are integers.

Sample Input 1

1

Sample Output 1

1

The only binary tree that satisfies the condition is a tree with one vertex whose root has $1$ written on it.

Sample Input 2

2

Sample Output 2

6

The binary trees that satisfy the condition are the six trees shown below.

Sample Input 3

222

Sample Output 3

987355927

Sample Input 4

222222

Sample Output 4

675337738