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 Ruby (2.7.1) (judge verdict: Time Limit Exceeded)

END { load __FILE__ unless $stdin.eof? }

class FenwickTree
  def initialize(n = 0)
    @n = n
    @data = Array.new(n, 0)
  end

  def add(idx, x)
    idx += 1
    while idx <= @n
      @data[idx - 1] += x
      @data[idx - 1] %= $m
      idx += idx & -idx
    end
    x
  end

  def sum(l, r)
    (__sum__(r) - __sum__(l)) % $m
  end

  private

  def __sum__(r)
    s = 0
    while r > 0
      s += @data[r - 1]
      r -= r & -r
    end
    s
  end
end

$m = 998244353
n, k = gets.split.map!(&:to_i)
ranges = Array.new(k) { gets.split.map!(&:to_i) }

table = FenwickTree.new(n + 1)
table.add(1, 1)

2.upto(n) do |i|
  ranges.each do |l, r|
    l, r = i - r, i - l
    l = 0 if l < 0
    r = 0 if r < 0
    next if r == 0
    table.add(i, table.sum(l, r + 1))
  end
end

puts table.sum(n, n + 1)