A14939. Basis

For an array of integers $a$ , let's define $|a|$ as the number of elements in it.

Let's denote two functions:

- $F(a, k)$ is a function that takes an array of integers $a$ and a positive integer $k$ . The result of this function is the array containing $|a|$ first elements of the array that you get by replacing each element of $a$ with exactly $k$ copies of that element.For example, $F([2, 2, 1, 3, 5, 6, 8], 2)$ is calculated as follows: first, you replace each element of the array with $2$ copies of it, so you obtain $[2, 2, 2, 2, 1, 1, 3, 3, 5, 5, 6, 6, 8, 8]$ . Then, you take the first $7$ elements of the array you obtained, so the result of the function is $[2, 2, 2, 2, 1, 1, 3]$ .
- $G(a, x, y)$ is a function that takes an array of integers $a$ and two different integers $x$ and $y$ . The result of this function is the array $a$ with every element equal to $x$ replaced by $y$ , and every element equal to $y$ replaced by $x$ .For example, $G([1, 1, 2, 3, 5], 3, 1) = [3, 3, 2, 1, 5]$ .

An array $a$ is a parent of the array $b$ if:

- either there exists a positive integer $k$ such that $F(a, k) = b$ ;
- or there exist two different integers $x$ and $y$ such that $G(a, x, y) = b$ .

An array $a$ is an ancestor of the array $b$ if there exists a finite sequence of arrays $c_0, c_1, \dots, c_m$ ( $m \ge 0$ ) such that $c_0$ is $a$ , $c_m$ is $b$ , and for every $i \in [1, m]$ , $c_{i-1}$ is a parent of $c_i$ .

And now, the problem itself.

You are given two integers $n$ and $k$ . Your goal is to construct a sequence of arrays $s_1, s_2, \dots, s_m$ in such a way that:

- every array $s_i$ contains exactly $n$ elements, and all elements are integers from $1$ to $k$ ;
- for every array $a$ consisting of exactly $n$ integers from $1$ to $k$ , the sequence contains at least one array $s_i$ such that $s_i$ is an ancestor of $a$ .

Print the minimum number of arrays in such sequence.

The only line contains two integers $n$ and $k$ ( $1 \le n, k \le 2 \cdot 10^5$ ).

Print one integer — the minimum number of elements in a sequence of arrays meeting the constraints. Since the answer can be large, output it modulo $998244353$ .

Let's analyze the first example.

One of the possible answers for the first example is the sequence $[[2, 1, 2], [1, 2, 2]]$ . Every array of size $3$ consisting of elements from $1$ to $2$ has an ancestor in this sequence:

- for the array $[1, 1, 1]$ , the ancestor is $[1, 2, 2]$ : $F([1, 2, 2], 13) = [1, 1, 1]$ ;
- for the array $[1, 1, 2]$ , the ancestor is $[1, 2, 2]$ : $F([1, 2, 2], 2) = [1, 1, 2]$ ;
- for the array $[1, 2, 1]$ , the ancestor is $[2, 1, 2]$ : $G([2, 1, 2], 1, 2) = [1, 2, 1]$ ;
- for the array $[1, 2, 2]$ , the ancestor is $[1, 2, 2]$ ;
- for the array $[2, 1, 1]$ , the ancestor is $[1, 2, 2]$ : $G([1, 2, 2], 1, 2) = [2, 1, 1]$ ;
- for the array $[2, 1, 2]$ , the ancestor is $[2, 1, 2]$ ;
- for the array $[2, 2, 1]$ , the ancestor is $[2, 1, 2]$ : $F([2, 1, 2], 2) = [2, 2, 1]$ ;
- for the array $[2, 2, 2]$ , the ancestor is $[1, 2, 2]$ : $G(F([1, 2, 2], 4), 1, 2) = G([1, 1, 1], 1, 2) = [2, 2, 2]$ .