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 JavaScript (Node.js 12.16.1) (judge verdict: Time Limit Exceeded)

const main = (input) => {
  const mod = 998244353n
  const [f, ...q] = input.trim().split('\n')
  const [n,] = f.split(' ').map(Number)

  const _s = {}
  for (let i=0; i<q.length; i++) {
    const [a, b] = q[i].split(' ').map(Number)

    for (let x=a; x<=b; x++) _s[x]++
  }

  const s = Array.from(Object.keys(_s)).sort((a, b) => a - b)

  const arr = new Array(n+1).fill(0n)
  arr[1] = 1n

  for (let i=1; i<n; i++) {
    if (arr[i] === 0n) continue

    for (let x=0; x<s.length; x++) {
      const p = Number(s[x])

      if (arr[i+p] !== undefined) arr[i+p] += arr[i]
      else break
    }
  }

  return console.log(`${(arr[n] % mod + mod) % mod}`)
}

process.env.MYTEST
  ? (process.env.MYTEST === 'test' ? test() : main(require('fs').readFileSync('dev/stdin', 'utf8')))
  : main(require('fs').readFileSync('/dev/stdin', 'utf8'))