P15707 [JAG 2023 Summer Camp #2] Mercurialist

This country has a medicine for immortality. Alice got $X + Y + Z$ bottles from the Hatter. $X$ bottles contain *elixir*. If Alice drinks it, she will immediately become immortal. $Y$ bottles contain mercury, and each has a different toxicity. If she drinks the $i$-th bottle, the following event $i$ will occur after $K + i - 0.5$ days. * Event $i$: Alice will immediately die if she has not drunk the elixir before event $i$. If she has drunk the elixir, she won't die. The remaining $Z$ bottles contain yogurt. Nothing will happen when Alice drinks it. At the same time every morning, Alice chooses one non-empty bottle with equal probability and drinks it. If all bottles are empty, she does nothing. Answer the probability that Alice will be alive $10^{10^{10}}$ days after the first day she starts drinking bottles. Note that Alice won't die other than events. The probability can be expressed as $\frac{P}{Q}$ using coprime integers $P$ and $Q$. Output a non-negative integer $R$ less than $998244353$ such that $R \times Q \equiv P \pmod{998244353}$. It can be proven that the probability is a rational number, and $R$ is uniquely determined under the conditions of this problem.

$X \ Y \ Z \ K$ The input satisfies the following constraints. * All inputs consist of integers. * $1 \le X, Y, Z, K \le 10^5$

Output $R$ defined in the statement. Add a new line at the end of the output.

In Sample Input 1, Alice will only die if she drinks mercury on day $1$ and yogurt on day $2$. The probability of death is $1/3 \times 1/2 = 1/6$, therefore the answer is $5/6$. In Sample Input 2, Alice never dies.