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No.2685 Cell Proliferation (Easy)

レベル : / AI推定難易度 : 1300 / 実行時間制限 : 1ケース 2.000秒 / メモリ制限 : 512 MB / 標準ジャッジ問題
タグ : （解説公開後に AC するまで非表示） / 解いたユーザー数 75
作問者 : ma_tw / テスター : 👑 tute7627 SPD_9X2 👑 rin204 だれ Ayuna
お気に入りにしたユーザー ProblemId : 10444 / 自分の提出
問題文最終更新日: 2024-03-21 00:41:44
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yukicoder contest 422 (第１回 競技プログラミング講習会作問企画コンテスト) (順位表) の他の問題:

Problem Statement

This problem differs from Cell Proliferation (Hard) only in the constraints and the execution time limit.

An AIC cell performs the following actions for $t=1,2,...$.
  • When $t$ seconds have passed since its birth, it immediately gives birth to $1$ new AIC cell with probability $\displaystyle\frac{P_1}{P_2}$.
  • When $(t + 0.5)$ seconds have passed since its birth, it immediately dies and disappears with probability $\displaystyle1 - \left(\frac{Q_1}{Q_2}\right)^t$.
The number of AIC cells does not change except through the actions above.
Now, exactly $1$ AIC cell has just been born. Find the expected value, $\mathrm{mod} \ 998244353$, of the number of AIC cells at the time $(T + 0.9)$ seconds after the current time.
Definition of the expected value $\mathrm{mod} \ 998244353$
It can be proved that the expected value to be found is always a rational number. Also, under the constraints of this problem, it is guaranteed that when the expected value is expressed as an irreducible fraction $\displaystyle\frac{P}{Q}$, $Q$ is not divisible by $998244353$. In this case, there exists a unique integer $R$ such that $0 \le R < 998244353$ and $P \equiv QR \pmod{998244353}$. Output this $R$.

Input

$P_1 \ P_2 \ Q_1 \ Q_2 \ T$
  • $0 \le P_1 \le P_2 < 998244353, \ P_2 \ne 0$
  • $0 \le Q_1 \le Q_2 < 998244353, \ Q_2 \ne 0$
  • $1 \le T \le 10^3$
  • All input values are integers.

Output

Output the answer as described in "Definition of the expected value $\mathrm{mod} \ 998244353$" above.

Sample

Sample 1
Input
1 2 3 4 2
Output
577110018

The expected value at $2.9$ seconds after the current time is $\displaystyle\frac{91}{64}$. Since $91 \equiv 64 \times 577110018 \pmod{998244353}$, output $577110018$.

Sample 2
Input
1 998244352 12345678 23456789 1000
Output
687546754

Sample 3
Input
0 1858 0 2024 166
Output
0

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