100 atcoder#AGC019F. [AGC019F] Yes or No

[AGC019F] Yes or No

Score : 20002000 points

Problem Statement

You are participating in a quiz with N+MN + M questions and Yes/No answers.

It's known in advance that there are NN questions with answer Yes and MM questions with answer No, but the questions are given to you in random order.

You have no idea about correct answers to any of the questions. You answer questions one by one, and for each question you answer, you get to know the correct answer immediately after answering.

Suppose you follow a strategy maximizing the expected number of correct answers you give.

Let this expected number be P/QP/Q, an irreducible fraction. Let M=998244353M = 998244353. It can be proven that a unique integer RR between 00 and M1M - 1 exists such that P=Q×RP = Q \times R modulo MM, and it is equal to P×Q1P \times Q^{-1} modulo MM, where Q1Q^{-1} is the modular inverse of QQ. Find RR.

Constraints

  • 1N,M500,0001 \leq N, M \leq 500,000
  • Both NN and MM are integers.

Partial Score

  • 15001500 points will be awarded for passing the testset satisfying N=MN = M and 1N,M1051 \leq N, M \leq 10^5.

Input

Input is given from Standard Input in the following format:

NN MM

Output

Let P/QP/Q be the expected number of correct answers you give if you follow an optimal strategy, represented as an irreducible fraction. Print P×Q1P \times Q^{-1} modulo 998244353998244353.

1 1
499122178

There are two questions. You may answer randomly to the first question, and you'll succeed with 50% probability. Then, since you know the second answer is different from the first one, you'll succeed with 100% probability.

The expected number of your correct answers is 33 / 22. Thus, P=3P = 3, Q=2Q = 2, Q1=499122177Q^{-1} = 499122177 (modulo 998244353998244353), and P×Q1=499122178P \times Q^{-1} = 499122178 (again, modulo 998244353998244353).

2 2
831870297

The expected number of your correct answers is 1717 / 66.

3 4
770074220

The expected number of your correct answers is 169169 / 3535.

10 10
208827570
42 23
362936761