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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;

const int mod = 998244353;

int64_t mul(int64_t x, int64_t y) {
    return (x % mod) * (y % mod) % mod;
}

int fib(long long n, int64_t a = 0, int64_t b = 1, int64_t c = 1, int a0 = 0, int a1 = 1) {
    if (n == 0) return a0;
    int na0 = n & 1 ? (a * a0 + b * a1) % mod : a0;
    int na1 = n & 1 ? (b * a0 + c * a1) % mod : a1;
    return fib(n >> 1, (a * a + b * b) % mod, b * (a + c) % mod, (b * b + c * c) % mod, na0, na1);
}

int main() {
    int n;
    int64_t m;
    cin >> n >> m;
    vector<int> inv(1010101);
    inv[1] = 1;
    for (int i = 2; i < 1010101; i++) inv[i] = mul(inv[mod % i], mod - mod / i);
    int ans = fib(m + 2 * (n - 1));
    int C = 1;
    for (int i = 0; i <= n - 2; i++) {
        ans += mul(C, mod - fib(n * 2 - 2 - 2 * i));
        ans %= mod;
        C = mul(C, mul(m - 1 + i + 1, inv[i + 1]));
    }
    cout << ans << endl;
}