Score : $500$ points

问题陈述

你有一个整数 $1$ 和一枚骰子，骰子掷出的整数在 $1$ 到 $6$（包括两端点）之间，并且每个整数出现的概率相等。
当你手中的整数严格小于 $N$ 时，重复执行以下操作：

• 投掷骰子。若骰子显示数字 $x$，则将你的整数乘以 $x$。

求当你的整数最终变为 $N$ 时的概率，该概率对 $998244353$ 取模。

如何计算模 $998244353$ 下的概率？我们能够证明所求概率总是有理数。另外，在本题约束条件下，当该概率表示为两个互质整数 $P$ 和 $Q$ 的形式 $\frac{P}{Q}$ 时，我们可以证明存在一个唯一的整数 $R$，满足 $R \times Q \equiv P\pmod{998244353}$ 以及 $0 \leq R \lt 998244353$。找出这个 $R$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You have an integer $1$ and a die that shows integers between $1$ and $6$ (inclusive) with equal probability.
You repeat the following operation while your integer is strictly less than $N$:

• Cast a die. If it shows $x$, multiply your integer by $x$.

Find the probability, modulo $998244353$, that your integer ends up being $N$.

How to find a probability modulo $998244353$? We can prove that the sought probability is always rational. Additionally, under the constraints of this problem, when that value is represented as $\frac{P}{Q}$ with two coprime integers $P$ and $Q$, we can prove that there is a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Find this $R$.

Constraints

• $2 \leq N \leq 10^{18}$
• $N$ is an integer.

Input

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

$N$

Output

Print the probability, modulo $998244353$, that your integer ends up being $N$.

Sample Input 1

6

Sample Output 1

239578645

One of the possible procedures is as follows.

• Initially, your integer is $1$.
• You cast a die, and it shows $2$. Your integer becomes $1 \times 2 = 2$.
• You cast a die, and it shows $4$. Your integer becomes $2 \times 4 = 8$.
• Now your integer is not less than $6$, so you terminate the procedure.

Your integer ends up being $8$, which is not equal to $N = 6$.

The probability that your integer ends up being $6$ is $\frac{7}{25}$. Since $239578645 \times 25 \equiv 7 \pmod{998244353}$, print $239578645$.

Sample Input 2

7

Sample Output 2

0

No matter what the die shows, your integer never ends up being $7$.

Sample Input 3

300

Sample Output 3

183676961

Sample Input 4

979552051200000000

Sample Output 4

812376310

update @ 2024/3/10 08:24:38