Problem:
Score : 400 points

Problem Statement

There are N cells arranged in a row, numbered 1, 2, \ldots, N from left to right.

Tak lives in these cells and is currently on Cell 1. He is trying to reach Cell N by using the procedure described below.

You are given an integer K that is less than or equal to 10, and K non-intersecting segments [L_1, R_1], [L_2, R_2], \ldots, [L_K, R_K].
Let S be the union of these K segments.
Here, the segment [l, r] denotes the set consisting of all integers i that satisfy l \leq i \leq r.

When you are on Cell i, pick an integer d from S and move to Cell i + d. You cannot move out of the cells.

To help Tak, find the number of ways to go to Cell N, modulo 998244353.

Constraints

2 \leq N \leq 2 \times 10^5

1 \leq K \leq \min(N, 10)

1 \leq L_i \leq R_i \leq N

[L_i, R_i] and [L_j, R_j] do not intersect (i \neq j)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
L_1 R_1
L_2 R_2
:
L_K R_K

Output

Print the number of ways for Tak to go from Cell 1 to Cell N, modulo 998244353.

Sample Input 1

5 2
1 1
3 4

Sample Output 1

4

The set S is the union of the segment [1, 1] and the segment [3, 4], therefore S = \{ 1, 3, 4 \} holds.

There are 4 possible ways to get to Cell 5:

1 \to 2 \to 3 \to 4 \to 5,

1 \to 2 \to 5,

1 \to 4 \to 5 and

1 \to 5.

Sample Input 2

5 2
3 3
5 5

Sample Output 2

0

Because S = \{ 3, 5 \} holds, you cannot reach to Cell 5.
Print 0.

Sample Input 3

5 1
1 2

Sample Output 3

5

Sample Input 4

60 3
5 8
1 3
10 15

Sample Output 4

221823067

Note that you have to print the answer modulo 998244353.

Solution in Fortran (GNU Fortran 9.2.1) (judge verdict: Accepted)

program main
    use,intrinsic :: iso_fortran_env
    implicit none
    integer(int64):: n,k,i,j, ans=0
    integer(int64):: md = 998244353
    integer(int64), allocatable:: dp(:),dpsum(:),l(:),r(:)

    read*, n,k
    allocate(dp,source=[(merge(1_8,0_8,i==1),i=1,n)])
    allocate(dpsum(0:n),source=[(merge(1_8,0_8,i==1),i=0,n)])
    allocate(l(k),r(k))

    do i=1,k
        read*, l(i),r(i)
    end do

    do i=2,n
        do j=1,k; block; integer(int64):: ll,rr
            ll = i-r(j); rr=i-l(j)
            if (rr < 1) cycle
            ll = max(ll,1_8)
            dp(i)=mod(dp(i)+dpsum(rr)-dpsum(ll-1),md)
        end block; end do
        ! print'(*(i0,1x))', dp(:)
        dpsum(i)=dpsum(i-1)+dp(i)
    end do
    print'(i0)', dp(n)
end program main