Critical Hit

AtCoder

IDabc280_e

Time2000ms

Memory256MB

Difficulty—

There is a monster with initial stamina $N$. Takahashi repeatedly attacks the monster while the monster's stamina remains $1$ or greater. An attack by Takahashi reduces the monster's stamina by $2$ with probability $\frac{P}{100}$ and by $1$ with probability $1-\frac{P}{100}$. Find the expected value, modulo $998244353$ (see Notes), of the number of attacks before the monster's stamina becomes $0$ or less. ## Constraints * $1 \leq N \leq 2\times 10^5$ * $0 \leq P \leq 100$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $P$ [samples] ## Notes We can prove that the sought expected value is always a finite rational number. Moreover, under the Constraints of this problem, when the value is represented as $\frac{P}{Q}$ by two coprime integers $P$ and $Q$, we can show that there exists a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Print such $R$.

Samples

Input #1

3 10

Output #1

229596204

An attack by Takahashi reduces the monster's stamina by $2$ with probability $\frac{10}{100}=\frac{1}{10}$ and by $1$ with probability $\frac{100-10}{100}=\frac{9}{10}$.

*   The monster's initial stamina is $3$.
*   After the first attack, the monster's stamina is $2$ with probability $\frac{9}{10}$ and $1$ with probability $\frac{1}{10}$.
*   After the second attack, the monster's stamina is $1$ with probability $\frac{81}{100}$, $0$ with probability $\frac{18}{100}$, and $-1$ with probability $\frac{1}{100}$. With probability $\frac{18}{100}+\frac{1}{100}=\frac{19}{100}$, the stamina becomes $0$ or less, and Takahashi stops attacking after two attacks.
*   If the stamina remains $1$ after two attacks, the monster's stamina always becomes $0$ or less by the third attack, so he stops attacking after three attacks.

Therefore, the expected value is $2\times \frac{19}{100}+3\times\left(1-\frac{19}{100}\right)=\frac{281}{100}$. Since $229596204 \times 100 \equiv 281\pmod{998244353}$, print $229596204$.

Input #2

5 100

Output #2

3

Takahashi's attack always reduces the monster's stamina by $2$. After the second attack, the stamina remains $5-2\times 2=1$, so the third one is required.

Input #3

280 59

Output #3

567484387

API Response (JSON)

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      "statement_type": "Markdown",
      "content": "There is a monster with initial stamina $N$.  \nTakahashi repeatedly attacks the monster while the monster's stamina remains $1$ or greater.\nAn attack by Takahashi reduces the monster's stamina by $2$ ...",
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      "language": "English"
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