Score : $600$ points

问题描述

Takahashi 拥有一枚无偏向的六面骰子和一个小于 $10^9$ 的正整数 $R$。每次投掷骰子时，它会独立且等概率地显示出数字 $1,2,3,4,5,6$，与其他试验的结果无关。

Takahashi 将执行以下步骤。最初，$C=0$。

1. 投掷骰子并将 $C$ 增加 $1$。
2. 让 $X$ 为目前为止所显示数字的总和。如果 $X-R$ 是 $10^9$ 的倍数，则结束该过程。
3. 返回步骤 1。

求在该过程结束时 $C$ 的期望值，结果对 $998244353$ 取模。

以上为通义千问 qwen-max 翻译，仅供参考。

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.

Sample Input 1

1

Sample Output 1

291034221

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

Sample Input 2

720357616

Sample Output 2

153778832

update @ 2024/3/10 12:25:22