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Ex - Alchemy

Score : $600$ points

Problem Statement

Takahashi has $A$ kinds of level-$1$ gems, and $10^{10^{100}}$ gems of each of those kinds.
For an integer $n$ greater than or equal to $2$, he can put $n$ gems that satisfy all of the following conditions into a cauldron to generate a level-$n$ gem in return.

• No two gems are of the same kind.
• Every gem's level is less than $n$.
• For every integer $x$ greater than or equal to $2$, there is at most one level-$x$ gem.

Find the number of kinds of level-$N$ gems that Takahashi can obtain, modulo $998244353$.

Here, two level-$2$ or higher gems are considered to be of the same kind if and only if they are generated from the same set of gems.

• Two sets of gems are distinguished if and only if there is a gem in one of those sets such that the other set does not contain a gem of the same kind.

Any level-$1$ gem and any level-$2$ or higher gem are of different kinds.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq A \leq 10^9$
• $N$ and $A$ are integers.

Input

The input is given from Standard Input in the following format:

$N$ $A$

Output

Print the answer.

3 3

10

Here are ten ways to obtain a level-$3$ gem.

• Put three kinds of level-$1$ gems into the cauldron.
  • Takahashi has three kinds of level-$1$ gems, so there is one way to choose three kinds of level-$1$ gems. Thus, he can obtain one kind of level-$3$ gem in this way.
• Put one kind of level-$2$ gem and two kinds of level-$1$ gems into the cauldron.
  • A level-$2$ gem can be obtained by putting two kinds of level-$1$ gems into the cauldron.
    • Takahashi has three kinds of level-$1$ gems, so there are three ways to choose two kinds of level-$1$ gems. Thus, three kinds of level-$2$ gems are available here.
  • There are three kinds of level-$2$ gems, and three ways to choose two kinds of level-$1$ gems, so he can obtain $3 \times 3 = 9$ kinds of level-$3$ gems in this way.

1 100

100

200000 1000000000

797585162