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F - Hanjo 2

Score : $600$ points

Problem Statement

We have a rectangular room that is $H$ meters long and $W$ meters wide.
We will fill this entire room with tatami (rectangular mats) that are $2$ meters long and $1$ meter wide, and hanjo (square mats) that are $1$ meter long and $1$ meter wide. Each tatami can be placed vertically or horizontally.
How many ways are there to fill the room?
We distinguish ways that match only after rotation or reflection.

Since the count can be enormous, find it modulo $998244353$.

Constraints

• $1 \leq H \leq 6$
• $1 \leq W \leq 10^{12}$

Input

Input is given from Standard Input in the following format:

$H$ $W$

Output

Print the answer.

2 2

7

We have the following seven ways:

3 3

131

5 100

379944232

Be sure to find the count modulo $998244353$.