Coprime Summation Challenge

The adorable Theresa needs your help to compute the following expression:

$$ S = \sum_{x = 1}^n\sum_{y = 1}^n\Bigg(\Bigl[ x \bot k_1\Bigr]\sum_{i = 1}^x\Bigl([i\bot x]\cdot i\Bigr)\cdot \Bigl[ y \bot k_2\Bigr]\sum_{j = 1}^y\Bigl([j \bot y]\cdot j\Bigr)\Bigg) $$

Here, the notation is defined as follows:

$$ [a \bot b] = \begin{cases} 1 & \text{if } \gcd(a,b)=1, \\ 0 & \text{otherwise} \end{cases} $$

You are given three positive integers: n, k1, and k2. You need to compute the sum S modulo 998244353.

Observe that the expression can be factored as follows:

$$ S = \Biggl(\sum_{x = 1}^n \Bigl[ x \bot k_1\Bigr] \cdot \left(\sum_{i = 1}^x [i\bot x] \cdot i\right)\Biggr) \times \Biggl(\sum_{y = 1}^n \Bigl[ y \bot k_2\Bigr] \cdot \left(\sum_{j = 1}^y [j\bot y] \cdot j\right)\Biggr) $$

Your task is to compute S mod 998244353.

inputFormat

The input consists of a single line containing three space-separated integers n, k1, and k2.

Constraints: You may assume that n is small enough (for example, n ≤ 1000) so that a simple solution works within the time limits.

outputFormat

Output a single integer: the value of S modulo 998244353.

sample

1 1 1

1