Description

After a long battle, Amazing John found a way to defeat the pufferfish:

There are $n$ “pufferfish” with “Divine Shield” and $m$ pufferfish without Divine Shield. Each time, with equal probability, you deal “Poison” damage to one surviving pufferfish.

Now Amazing John wants to know: what is the expected number of times you need to deal damage to kill all the pufferfish?

Output the answer modulo $998244353$.

“Divine Shield”: When a pufferfish with Divine Shield takes damage, it is immune to this instance of damage. After preventing the damage, the Divine Shield is destroyed.

“Pufferfish”: After a pufferfish’s Divine Shield is destroyed, it grants Divine Shield to all other pufferfish that are not dead and do not have Divine Shield.

“Poison”: Immediately kills a pufferfish without Divine Shield.

Input Format

The input consists of one line containing two non-negative integers $n,m$, meaning there are $n$ pufferfish with Divine Shield and $m$ pufferfish without Divine Shield.

Output Format

Output one line containing one non-negative integer $s$, the expected number of damage instances modulo $998244353$.

Specifically, the answer can always be written as $\frac{p}{q}(p,q\in \mathbb{N},q\neq 0)$, and you need to output $p×q^{-1}$ modulo $998244353$.

2 1

8

10 10

499122389

10 0

65

2 0

5

Hint

This problem has $20$ test points, numbered from $1$ to $20$. For a subtask, you can get the score of that subtask only if you pass all the test points in it.

The fraction form $\frac{p}{q}$ must satisfy $(p,q\in \mathbb{N},998244353\nmid q)$.

I will secretly support you, but do not tell anyone else——Bob.

Translated by ChatGPT 5