Problem:
Max Score: $1400$ Points

Problem Statement

E869120 defined a sequence $a$ like this:

$a_1=a_2=1$, $a_{k+2}=a_{k+1}+a_k \ (k \ge 1)$

He also defined sequences $d_1, d_2, d_3, \dots , d_n$, as the following recurrence relation :

$d_{1, j} = a_j$

$d_{i, j} = \sum_{k = 1}^j d_{i - 1, k} \ (i \ge 2)$

You are given integers $n$ and $m$. Please calculate the value of $d_{n, m}$.

Since the answer can be large number, print the answer modulo $998,244,353$.

Can you solve this problem???

Input

The input is given from standard input in the following format.

$n \quad m$

Output

Print $d_{n, m}$ modulo $998,244,353$.

Constraints

$1 \le n \le 200,000$

$1 \le m \le 10^{18}$

Subtasks

Subtask 1 [ $100$ points ]

The testcase in this subtask satisfies $1 \le n, m \le 3,000$.

Subtask 2 [ $170$ points ]

The testcase in this subtask satisfies $1 \le m \le 200,000$.

Subtask 3 [ $230$ points ]

The testcase in this subtask satisfies $1 \le n \le 3$.

Subtask 4 [ $420$ points ]

The testcase in this subtask satisfies $1 \le n \le 1000$.

Subtask 5 [ $480$ points ]

There are no additional constraints.

Sample Input 1

4 7

Sample Output 1

176

The sequence is following:

$d_{i, 1}$

$d_{i, 2}$

$d_{i, 3}$

$d_{i, 4}$

$d_{i, 5}$

$d_{i, 6}$

$d_{i, 7}$

$d_1$

1

1

2

3

5

8

13

$d_2$

1

2

4

7

12

20

33

$d_3$

1

3

7

14

26

46

79

$d_4$

1

4

11

25

51

97

176

As a result, the answer is $d_{4, 7} = 176$.

Sample Input 2

12 20

Sample Output 2

174174144

Interesting number. It only contains $1$, $4$ and $7$.

Sample Input 3

16 30

Sample Output 3

102292850

You can calculate that $d_{16, 30} = 1,193,004,294,685$.

Please don't forget printing the answer mod $998,244,353$.

Writer：square1001

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Text;
using System.Collections.Generic;
class Solve{
    int N;
    long M;
    public Solve(){}
    StringBuilder sb;
    public static int Main(){
        new Solve().Run();
        return 0;
    }
    void Run(){
        sb = new StringBuilder();
        Read();
        Calc();
        Console.Write(sb.ToString());
    }
    void Calc(){
        long[] fib = new long[N];
        for(int i=0;i<N;i++){
            if(i == 0 || i == 1){
                fib[i] = 1;
            }
            else{
                fib[i] = (fib[i-1] + fib[i-2]) % Define.mod;
            }
        }
        if(M == 1){
            sb.Append(fib[N-1]+"\n");
            return;
        }
        Conv c = new Conv(N,M-2);
        long count = 0;
        for(int i=0;i<N;i++){
            count = (count + (c.C(i)*fib[N-1-i]%Define.mod)) % Define.mod;
        }
        sb.Append(count+"\n");
    }
    void Read(){
        string[] str = Console.ReadLine().Split(' ');
        M = Int64.Parse(str[0]);
        N = int.Parse(str[1]);
    }    
}
class Conv{
    public int n;
    public long m;
    public long[] c;
    public Conv(int n0,long m0){
        n = n0;
        m = m0;
        c = new long[n];
        for(int i=0;i<n;i++){
            if(i == 0){
                c[i] = 1;
            }
            else{
                c[i] = c[i-1]*Rev(i)%Define.mod*((m+i)%Define.mod)%Define.mod;
            }
        }
    }
    public long C(int N){
        return c[N];
    }
    long Rev(long N){
        return Pow(N,Define.mod-2);
    }
    long Pow(long N,long K){
        if(K == 0){
            return 1;
        }
        else if(K % 2 == 0){
            long t = Pow(N,K/2);
            return t*t%Define.mod;
        }
        else{
            return N*Pow(N,K-1)%Define.mod;
        }
    }
}
public static class Define{
    public const long mod = 998244353;
}