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 C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Collections;
using System.Collections.Specialized;
using System.Linq;
using System.Text;
using System.IO;
using System.Reflection;
using static System.Math;
using System.Numerics;
using System.Threading;
static class Program{
	const int mod=(int)1e9+7;
	const double eps=1e-11;
	static void Main(){
		Sc sc=new Sc();
		var s=sc.Ia;
		var a=new int[s[1]][];
		for(int i = 0;i<s[1];i++) {
			a[i]=sc.Ia;
		}
		var st=new St(s[0]+1);
		st.Add(1,1,1);
		for(int i = 1;i<s[0];i++) {
			var d=st.Get(i,i);
			for(int j = 0;j<s[1];j++) {
				if(a[j][0]+i>s[0]){continue;}
				st.Add(a[j][0]+i,Min(a[j][1]+i,s[0]),d);
			}
		}
		Console.WriteLine("{0}",st.Get(s[0],s[0]));
	}
}
public class St{
	private Mint[] dat,de;
	private readonly int[] ha;
	private int n,cnt;
	public St(int n){
		this.n=n;
		cnt=n<<1;
		dat=new Mint[cnt];
		de=new Mint[cnt];
		ha=new int[cnt];
		for(int i = cnt-1,j=n,k=1;i>0;i--) {
			if(i==j){j>>=1;k<<=1;}
			ha[i]=k;
		}
	}
	public void Add(int q,Mint b){
		q+=n;
		dat[q]+=b;
		while(q>1){
			dat[q>>=1]=dat[q<<1]+dat[(q<<1)+1];
		}
	}
	public void Add(int l,int r,Mint b){
		int ln=l+=n,rn=r+=n;
		while(l<=r){
			if((l&1)==1){
				de[l]+=b;
				l++;
			}
			if((r&1)==0){
				de[r]+=b;
				r--;
			}
			l>>=1;r>>=1;
		}
		while((ln>>=1)!=0){
			Fp(ln<<1);
			Fp((ln<<1)+1);
			dat[ln]=dat[ln<<1]+dat[(ln<<1)+1];
		}
		while((rn>>=1)!=0){
			Fp(rn<<1);
			Fp((rn<<1)+1);
			dat[rn]=dat[rn<<1]+dat[(rn<<1)+1];
		}
	}
	public Mint Get(int l,int r){
		int ln=l+=n,rn=r+=n;
		var sk=new Stack<int>();
		while((ln>>=1)!=0){sk.Push(ln);}
		while((rn>>=1)!=0){sk.Push(rn);}
		while(sk.Count>0){Fp(sk.Pop());}
		Mint x=0;
		while(l<=r){
			if((l&1)==1){
				Fp(l);
				x+=dat[l++];
			}
			if((r&1)==0){
				Fp(r);
				x+=dat[r--];
			}
			l>>=1;r>>=1;
		}
		return x;
	}
	private void Fp(int a){
		if((int)de[a]!=0){
			dat[a]+=de[a]*ha[a];
			if(a<<1<cnt){
				de[a<<1]+=de[a];
				de[(a<<1)+1]+=de[a];
			}
			de[a]=0;
		}
	}
}
public struct Mint{
	const int mod=998244353;
	private long d;
	public Mint(long d){this.d=d;}
	public static implicit operator Mint(long d){return new Mint(d%mod);}
	public static explicit operator long(Mint d){return d.d;}
	public override string ToString(){return d.ToString();}
	public static Mint operator+(Mint a,long b){a.d=(a.d+b)%mod;return a;}
	public static Mint operator+(Mint a,Mint b){a.d=(a.d+b.d)%mod;return a;}
	public static Mint operator-(Mint a,long b){a.d=(mod+a.d-b)%mod;return a;}
	public static Mint operator-(Mint a,Mint b){a.d=(mod+a.d-b.d)%mod;return a;}
	public static Mint operator*(Mint a,long b){a.d=(a.d*b)%mod;return a;}
	public static Mint operator*(Mint a,Mint b){a.d=(a.d*b.d)%mod;return a;}
	public static Mint operator/(Mint a,long b){a.d=(a.d*Mi(b))%mod;return a;}
	public static Mint operator/(Mint a,Mint b){a.d=(a.d*Mi(b.d))%mod;return a;}
	private static long Mi(long a){
		a=(a+mod)%mod;
		long b=mod,u=1,v=0;
		while(b>0){
			long t=a/b;
			a-=t*b;(a,b)=(b,a);
			u-=t*v;(u,v)=(v,u);
		}
		u%=mod;
		if(u<0){u+=mod;}
		return u%mod;
	}
}
public class Sc{
	public int I{get{return int.Parse(Console.ReadLine());}}
	public long L{get{return long.Parse(Console.ReadLine());}}
	public double D{get{return double.Parse(Console.ReadLine());}}
	public string S{get{return Console.ReadLine();}}
	public int[] Ia{get{return Array.ConvertAll(Console.ReadLine().Split(),int.Parse);}}
	public long[] La{get{return Array.ConvertAll(Console.ReadLine().Split(),long.Parse);}}
	public double[] Da{get{return Array.ConvertAll(Console.ReadLine().Split(),double.Parse);}}
	public string[] Sa{get{return Console.ReadLine().Split();}}
	public object[] Oa{get{return Console.ReadLine().Split();}}
	public int[] Ia2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),int.Parse);}}
	public int[] Ia3(string a,string b){return Array.ConvertAll((a+Console.ReadLine()+b).Split(),int.Parse);}
	public int[] Ia3(int a){return Array.ConvertAll((Console.ReadLine()+" "+a.ToString()).Split(),int.Parse);}
	public long[] La2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),long.Parse);}}
	public long[] La3(string a,string b){return Array.ConvertAll((a+Console.ReadLine()+b).Split(),long.Parse);}
	public long[] La3(int a){return Array.ConvertAll((Console.ReadLine()+" "+a.ToString()).Split(),long.Parse);}
	public double[] Da2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),double.Parse);}}
	public double[] Da3(string a,string b){return Array.ConvertAll((a+Console.ReadLine()+b).Split(),double.Parse);}
	public T[] Arr<T>(int n,Func<T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f();}return a;}
	public T[] Arr<T>(int n,Func<int,T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i);}return a;}
	public T[] Arr<T>(int n,Func<string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(Console.ReadLine().Split());}return a;}
	public T[] Arr<T>(int n,Func<int,string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i,Console.ReadLine().Split());}return a;}
}