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G - Count Sequences

Score : $600$ points

Problem Statement

Find the number of sequences of integers with $N$ terms, $A=(a_1,a_2,\ldots,a_N)$, that satisfy the following conditions, modulo $998244353$.

• $0 \leq a_1 \leq a_2 \leq \ldots \leq a_N \leq M$.
• For each $i=1,2,\ldots,N-1$, the remainder when $a_i$ is divided by $3$ is different from the remainder when $a_{i+1}$ is divided by $3$.

Constraints

• $2 \leq N \leq 10^7$
• $1 \leq M \leq 10^7$
• All values in the input are integers.

Input

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

$N$ $M$

Output

Print the answer.

3 4

8

Here are the eight sequences that satisfy the conditions.

• $(0,1,2)$
• $(0,1,3)$
• $(0,2,3)$
• $(0,2,4)$
• $(1,2,3)$
• $(1,2,4)$
• $(1,3,4)$
• $(2,3,4)$

276 10000000

909213205

Be sure to find the count modulo $998244353$.