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 Julia (1.4.0) (judge verdict: Wrong Answer)

function ABC_D()
    readints()=parse.(Int,split(readline()))
    MOD=998244353
    N,K=readints()

    L=zeros(Int,K)
    R=zeros(Int,K)
    for i in 1:K
        L[i],R[i]=readints()
    end
    S0=Set(hcat(L,R))
    S=sort!(collect(S0))


    NMax = 2*10^5
    fac=ones(Int,NMax)#factorial_mod
    for i in 2:NMax
        fac[i]=fac[i-1]*i%MOD
    end

    function CMOD(a,b)#binominal_mod
        fac[a]*invmod(fac[b],MOD)%MOD*invmod(fac[a-b],MOD)%MOD
    end
    N1=N-1

    nk=zeros(Int,length(S))
    function nokori(nk)
        N1-(sum(S .* nk)-S[1]*nk[1])
    end
    function calcP(nk)

        ret=fac[sum(nk)]
        for i in 1:length(nk)
            if nk[i]==0
                im =  1
            else
                im= invmod(fac[nk[i]],MOD)
            end
            ret*=im
            ret%=MOD
        end
        ret
    end

    function S_plus(nk)
        nk[end]+=1
        for i in length(nk):-1:3
            if nk[i]>Smax[i]
                nk[i]=0
                nk[i-1]+=1
            end
        end
        if length(nk)>1
            if nk[2]>Smax[end]
                return false
            end
        else
            return false
        end
        true
    end       

    nk=zeros(Int,length(S))
    Smax=@. N1 ÷ S
    chk=0

    flag=true
    println(S)
    while flag
        NOKORI=nokori(nk)
        if NOKORI % S[1] == 0 && NOKORI ≥ 0
            nk[1]=NOKORI÷S[1]
            println("$nk = > $(sum(nk.*S))")
            chk+=calcP(nk)
            chk%=MOD
        end
        flag=S_plus(nk)
    end
    print(chk)
end
    
ABC_D()