Score: $800$ points

问题陈述

给定整数 $N$ 和 $P$。

存在一个包含 $N$ 个顶点和 $N$ 条边的图，其中每个顶点被标记为 $1$ 到 $N$。第 $i$ 条边双向连接顶点 $i$ 和 $i+1$。在这里，顶点 $N+1$ 指的是顶点 $1$。

执行以下算法以获得长度为 $N$ 的序列 $D=(D_1,D_2,\ldots,D_N)$：

• 将长度为 $N$ 的整数序列 $D$ 设置为 $D=(D_1,\ldots,D_N)=(-1,\ldots,-1)$。同时，将数字对的序列 $Q$ 设置为 $Q=((1,0))$。当 $Q$ 不为空时，重复以下过程：

  • 让 $(v,d)$ 是 $Q$ 的第一个元素。移除这个元素。

  • 如果 $D_v = -1$，那么设置 $D_v := d$，并且对于每个与顶点 $v$ 相邻且 $D_x=-1$ 的顶点 $x$，执行以下过程。如果有多个这样的 $x$ 满足条件，按照顶点编号的升序处理它们：

    1. 以概率 $\frac{P}{100}$，将 $(x,d+1)$ 添加到 $Q$ 的 前面。
    2. 如果 $(x,d+1)$ 没有被添加到 $Q$ 的前面，就将它添加到 $Q$ 的 后面。

找到最终获得的序列 $D$ 的元素之和的期望值，对 $998244353$ 取模。

解决给定的每个 $T$ 个测试用例。

期望值的定义 $\text{mod } 998244353$ 可以证明要找到的期望值总是一个有理数。此外，这个问题的约束条件保证如果该值表示为一个不可约分数 $\frac{P}{Q}$，那么 $Q$ 不能被 $998244353$ 整除。在这里，存在一个唯一的整数 $R$ 在 $0$ 和 $998244352$ 之间，包括 $0$ 和 $998244352$，使得 $R\times Q \equiv P\pmod{998244353}$。提供这个 $R$ 作为答案。

以上为大语言模型 kimi 翻译，仅供参考。

Problem Statement

You are given integers $N$ and $P$.

There is a graph with $N$ vertices and $N$ edges, where each vertex is labeled $1$ to $N$. The $i$-th edge connects vertices $i$ and $i+1$ bidirectionally. Here, vertex $N+1$ refers to vertex $1$.

Perform the following algorithm to obtain a sequence $D=(D_1,D_2,\ldots,D_N)$ of length $N$:

• Set an integer sequence $D$ of length $N$ to $D=(D_1,\ldots,D_N)=(-1,\ldots,-1)$. Also, set a sequence $Q$ of number pairs to $Q=((1,0))$. Repeat the following process while $Q$ is not empty:

  • Let $(v,d)$ be the first element of $Q$. Remove this element.

  • If $D_v = -1$, then set $D_v := d$, and for each vertex $x$ adjacent to vertex $v$ such that $D_x=-1$, perform the following process. If there are multiple such $x$ that satisfy the condition, process them in ascending order of vertex number:

    1. With probability $\frac{P}{100}$, add $(x,d+1)$ to the front of $Q$.
    2. If $(x,d+1)$ was not added to the front of $Q$, add it to the end of $Q$.

Find the expected value of the sum of the elements of the final sequence $D$ obtained, modulo $998244353$.

Solve each of the $T$ test cases given.

Definition of expected value $\text{mod } 998244353$ It can be proved that the expected value to be found is always a rational number. Furthermore, the constraints of this problem guarantee that if that value is expressed as an irreducible fraction $\frac{P}{Q}$, then $Q$ is not divisible by $998244353$. Here, there is a unique integer $R$ between $0$ and $998244352$, inclusive, such that $R\times Q \equiv P\pmod{998244353}$. Provide this $R$ as the answer.

Constraints

• $1 \leq T \leq 10^4$
• $3 \leq N \leq 10^{18}$
• $1\leq P \leq 99$
• All input numbers are integers.

Input

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

$T$

$\mathrm{case}_1$

$\vdots$

$\mathrm{case}_T$

Each case is given in the following format:

$N$ $P$

Output

Print $T$ lines. The $i$-th line $(1 \leq i \leq T)$ should contain the answer for the $i$-th test case.

Sample Input 1

3
3 50
4 1
1000000000000000000 70

Sample Output 1

499122179
595552585
760296751

In the first test case, the algorithm may operate as follows:

• Initially, $D=(-1,-1,-1)$ and $Q=((1,0))$. Remove the first element $(1,0)$ from $Q$.
• $D_1 = -1$, so set $D_1 := 0$. The vertices $x$ adjacent to vertex $1$ such that $D_x= -1$ are $2$ and $3$.
• Add $(2,1)$ to the front of $Q$. Add $(3,1)$ to the end of $Q$. Now $Q=((2,1),(3,1))$.
• Remove the first element $(2,1)$ from $Q$.
• $D_2 = -1$, so set $D_2 := 1$. The vertex $x$ adjacent to vertex $2$ such that $D_x= -1$ is $3$.
• Add $(3,2)$ to the front of $Q$. Now $Q=((3,2),(3,1))$.
• Remove the first element $(3,2)$ from $Q$.
• $D_3 = -1$, so set $D_3 := 2$. There are no vertices $x$ adjacent to vertex $3$ such that $D_x= -1$, so do nothing.
• Remove the first element $(3,1)$ from $Q$.
• $D_3 =2$, so do nothing.
• $Q$ is now empty, so the process ends.

In this case, the final sequence obtained is $D=(0,1,2)$. The probability that the algorithm operates as described above is $\frac{1}{8}$, and the expected sum of the elements of $D$ is $\frac{5}{2}$.