Score : $1800$ points

Problem Statement

We have an $N \times M$ grid. The square at the $i$-th row and $j$-th column will be denoted as $(i,j)$. Particularly, the top-left square will be denoted as $(1,1)$, and the bottom-right square will be denoted as $(N,M)$. Takahashi painted some of the squares (possibly zero) black, and painted the other squares white.

We will define an integer sequence $A$ of length $N$, and two integer sequences $B$ and $C$ of length $M$ each, as follows:

• $A_i(1\leq i\leq N)$ is the minimum $j$ such that $(i,j)$ is painted black, or $M+1$ if it does not exist.
• $B_i(1\leq i\leq M)$ is the minimum $k$ such that $(k,i)$ is painted black, or $N+1$ if it does not exist.
• $C_i(1\leq i\leq M)$ is the maximum $k$ such that $(k,i)$ is painted black, or $0$ if it does not exist.

How many triples $(A,B,C)$ can occur? Find the count modulo $998244353$.

Notice

In this problem, the length of your source code must be at most $20000$ B. Note that we will invalidate submissions that exceed the maximum length.

Constraints

• $1 \leq N \leq 8000$
• $1 \leq M \leq 200$
• $N$ and $M$ are integers.

Partial Score

• $1500$ points will be awarded for passing the test set satisfying $N\leq 300$.

Input

Input is given from Standard Input in the following format:

N M

Output

Print the number of triples $(A,B,C)$, modulo $998244353$.

2 3

64

Since $N=2$, given $B_i$ and $C_i$, we can uniquely determine the arrangement of black squares in each column. For each $i$, there are four possible pairs $(B_i,C_i)$: $(1,1)$, $(1,2)$, $(2,2)$ and $(3,0)$. Thus, the answer is $4^M=64$.

4 3

2588

17 13

229876268

5000 100

57613837