Ex - Dice Sum Infinity

AtCoder

IDabc299_h

Time2000ms

Memory256MB

Difficulty—

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 $998244353$. ## Constraints * $0\lt R\lt10^9$ * $R$ is an integer. ## Input The input is given from Standard Input in the following format: $R$ [samples] ## 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 $x$.

Samples

Input #1

Output #1

291034221

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

Input #2

720357616

Output #2

153778832

API Response (JSON)

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      "description_type": "Markdown"
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    "platform": "AtCoder",
    "limit": {
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      "content": "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 o...",
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