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 Python (3.8.2) (judge verdict: Time Limit Exceeded)

import sys
input = sys.stdin.readline
sys.setrecursionlimit(10**9)

def solve():
    MOD = 998244353

    def makeBIT(numEle):
        numPow2 = 2 ** (numEle-1).bit_length()
        data = [0] * (numPow2+1)
        return data, numPow2
    def setInit(As):
        for iB, A in enumerate(As, 1):
            data[iB] = A
        for iB in range(1, numPow2):
            i = iB + (iB & -iB)
            data[i] += data[iB]
    def addValue(iA, A):
        iB = iA + 1
        while iB <= numPow2:
            data[iB] += A
            iB += iB & -iB
    def getSum(iA):
        iB = iA + 1
        ans = 0
        while iB > 0:
            ans += data[iB]
            iB -= iB & -iB
        return ans
    def getRangeSum(iFr, iTo):
        return getSum(iTo) - getSum(iFr-1)


    N, K = map(int, input().split())
    LRs = [tuple(map(int, input().split())) for _ in range(K)]

    data, numPow2 = makeBIT(N)

    addValue(0, 1)

    for i in range(1, N):
        v = 0
        for L, R in LRs:
            x, y = i-R, i-L
            if y < 0:
                continue
            if x < 0:
                x = 0
            v += getRangeSum(x, y) % MOD
            v %= MOD
        addValue(i, v)

    ans = getRangeSum(N-1, N-1) % MOD
    print(ans)


solve()