Score : $600$ points

问题陈述

高桥决定在他的网页上放置一个网页计数器。
访问他网页的情况如下：

• 对于 $i=0,1,2,\ldots,23$，每天的 $i$ 点钟都有可能进行访问：
  • 如果 $c_i=$T，高桥访问该网页的概率为 $X$%。
  • 如果 $c_i=$A，青木访问该网页的概率为 $Y$%。
  • 高桥或青木是否访问网页在每次都是独立决定的。
• 没有其他访问来源。

另外，高桥认为自从放置计数器以来，第 $N$ 次访问最好是不由他自己发起的。

如果高桥在某天 恰好在 0 点前 放置计数器，请计算（模 $998244353$）第 $N$ 次访问是由青木发起的概率。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Takahashi has decided to put a web counter on his webpage.
The accesses to his webpage are described as follows:

• For $i=0,1,2,\ldots,23$, there is a possible access at $i$ o'clock every day:
  • If $c_i=$T, Takahashi accesses the webpage with a probability of $X$ percent.
  • If $c_i=$A, Aoki accesses the webpage with a probability of $Y$ percent.
  • Whether or not Takahashi or Aoki accesses the webpage is determined independently every time.
• There is no other access.

Also, Takahashi believes it is preferable that the $N$-th access since the counter is put is not made by Takahashi himself.

If Takahashi puts the counter right before $0$ o'clock of one day, find the probability, modulo $998244353$, that the $N$-th access is made by Aoki.

Notes

We can prove that the sought probability is always a finite rational number. Moreover, under the constraints of this problem, when the 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

• $1 \leq N \leq 10^{18}$
• $1 \leq X,Y \leq 99$
• $c_i$ is T or A.
• $N$, $X$, and $Y$ are integers.

Input

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

$N$ $X$ $Y$

$c_0 c_1 \ldots c_{23}$

Output

Print the answer.

Sample Input 1

1 50 50
ATATATATATATATATATATATAT

Sample Output 1

665496236

The $1$-st access since Takahashi puts the web counter is made by Aoki with a probability of $\frac{2}{3}$.

Sample Input 2

271 95 1
TTTTTTTTTTTTTTTTTTTTTTTT

Sample Output 2

0

There is no access by Aoki.

Sample Input 3

10000000000000000 62 20
ATAATTATATTTAAAATATTATAT

Sample Output 3

744124544

update @ 2024/3/10 11:26:52