24714번 - Blind Box 다국어

You are the owner of a store. 

The store has launched a new blind box campaign. Each blind box contains $n$ cards, and there is a positive integer written on each card. The cards in each box are ordered in such a way that the number on the $i$-th card is greater than or equal to the number on the $(i{-}1)$-th card for every $i > 1$. Additionally, the integer on each card does not exceed $m$.

The store has all possible blind boxes satisfying the conditions above, and every two blind boxes in the store are different. Two boxes are considered different if and only if there is an index $i$ such that the numbers on the $i$-th cards in the two boxes are different.

You sell blind boxes at a fixed price. After buying and opening a blind box, customers will ask you for a cashback, and the amount equals the product of the numbers on the $n$ cards in the box. Please calculate the minimum price of each blind box to ensure that, after selling all blind boxes, your net income is non-negative.

The first line of input contains two integers $n$ and $m$: the number of cards in each box and the maximum value on a card ($1 \leq n, m \leq 10^5$).

Print a single integer: the minimum price to ensure a non-negative net income. The price may be fractional, but you have to output this price modulo $998\,244\,353$. Formally, let the minimum price be an irreducible fraction $\frac{x}{y}$. They you have to print $x \cdot y^{-1} \bmod 998\,244\,353$, where $y^{-1}$ is an integer such that $y \cdot y^{-1} \bmod 998\,244\,353 = 1$.

Explanation of the first example:

There are three different blind boxes: $(1,1)$, $(1,2)$, and $(2,2)$.

The amounts of cashback are $1$, $2$, and $4$, respectively.

So, the minimum price should be $\frac{7}{3}$.

And the answer in the second example is $\frac{42525}{126} = \frac{675}{2}$.