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Ex - Dice Sum Infinity

Score : $600$ points

Problem Statement

Takahashi has an unbiased six-sided die and a positive integer $R$ less than $10^9$. Each time the die is cast, it shows one of the numbers $1,2,3,4,5,6$ with equal probability, independently of the outcomes of the other trials.

Takahashi will perform the following procedure. Initially, $C=0$.

1. Cast the die and increment $C$ by $1$.
2. Let $X$ be the sum of the numbers shown so far. If $X-R$ is a multiple of $10^9$, quit the procedure.
3. Go back to step 1.

Find the expected value of $C$ at the end of the procedure, modulo $998244353$.

Notes

Under the constraints of this problem, it can be shown that the expected value of $C$ is represented as an irreducible fraction $p/q$, and there is a unique integer $x\ (0\leq x\lt998244353)$ such that $xq \equiv p\pmod{998244353}$. Print this $x$.

Constraints

• $0\lt R\lt10^9$
• $R$ is an integer.

Input

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

$R$

Output

Print a single line containing the answer.

1

291034221

The expected value of $C$ at the end of the procedure is approximately $833333333.619047619$, and $291034221$ when represented modulo $998244353$.

720357616

153778832