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)

Devisor = 998244353
N, K = gets.chomp.split(' ').map(&:to_i)
sectinos = Array.new(K){ gets.chomp.split(' ').map(&:to_i) }

union = []
sectinos.each do |section|
    s, e = section
    s.upto(e) { |i| union.push(i) }
end

# lcm = union.inject(:lcm)

rest = N - 1
min = union.min
max = union.max

# ans = 0
# 1.upto(rest / min) do |k|
#     union.repeated_permutation(k) do |perm|
#         ans += 1 if perm.sum == rest
#     end
# end
# puts ans % Devisor

mas = Array.new(N + 1){ 0 }
union.each do |u|
    mas[-1 - u] += 1 if -1 - u >= N * -1
end
# p mas

(-1 - min).downto(N * -1 + min) do |i|
    next if mas[i] == 0
    union.each do |u|
        # puts "i = #{i}, u = #{u}"
        pos = i - u
        mas[pos] += mas[i] if pos >= N * -1
        # p mas
    end
end
puts mas[1] % Devisor