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: Accepted)

# Binary Indexed Tree
# https://www.topcoder.com/community/competitive-programming/tutorials/binary-indexed-trees
# http://hos.ac/slides/20140319_bit.pdf
class BIT
  def initialize(n)
    @tree = Array.new(n + 1, 0)
  end

  def add(i, diff)
    while i < @tree.size
      @tree[i] += diff
      i += (i & -i)
    end
  end

  def sum(i)
    t = 0
    while i > 0 do
      t += @tree[i]
      i -= (i & -i)
    end
    return t
  end
end

n, k = gets.chomp.split.map(&:to_i)

bit = BIT.new(n)
bit.add(1, 1)

rng = []
k.times do
  l, r = gets.chomp.split.map(&:to_i)
  rng.push([l, r])
end

2.upto(n) do |i|
  t = 0

  rng.each do |a, b|
    aa = i - a
    bb = i - b - 1

    aa = 0 if aa < 0
    bb = 0 if bb < 0

    t += bit.sum(aa) - bit.sum(bb)
  end

  bit.add(i, t % 998244353)
end

puts bit.sum(n) - bit.sum(n-1)