Swimmers in the Pool

3000ms

524288K

Description:

There is a pool of length $$$l$$$ where $$$n$$$ swimmers plan to swim. People start swimming at the same time (at the time moment $$$0$$$), but you can assume that they take different lanes, so they don't interfere with each other.

Each person swims along the following route: they start at point $$$0$$$ and swim to point $$$l$$$ with constant speed (which is equal to $$$v_i$$$ units per second for the $$$i$$$-th swimmer). After reaching the point $$$l$$$, the swimmer instantly (in negligible time) turns back and starts swimming to the point $$$0$$$ with the same constant speed. After returning to the point $$$0$$$, the swimmer starts swimming to the point $$$l$$$, and so on.

Let's say that some real moment of time is a meeting moment if there are at least two swimmers that are in the same point of the pool at that moment of time (that point may be $$$0$$$ or $$$l$$$ as well as any other real point inside the pool).

The pool will be open for $$$t$$$ seconds. You have to calculate the number of meeting moments while the pool is open. Since the answer may be very large, print it modulo $$$10^9 + 7$$$.

Input:

The first line contains two integers $$$l$$$ and $$$t$$$ ($$$1 \le l, t \le 10^9$$$) — the length of the pool and the duration of the process (in seconds).

The second line contains the single integer $$$n$$$ ($$$2 \le n \le 2 \cdot 10^5$$$) — the number of swimmers.

The third line contains $$$n$$$ integers $$$v_1, v_2, \dots, v_n$$$ ($$$1 \le v_i \le 2 \cdot 10^5$$$), where $$$v_i$$$ is the speed of the $$$i$$$-th swimmer. All $$$v_i$$$ are pairwise distinct.

Output:

Print one integer — the number of meeting moments (including moment $$$t$$$ if needed and excluding moment $$$0$$$), taken modulo $$$10^9 + 7$$$.

Sample Input:

9 18
2
1 2

Sample Output:

Sample Input:

12 13
3
4 2 6

Sample Output:

Sample Input:

1 1000000000
3
100000 150000 200000

Sample Output:

997200007

Note:

In the first example, there are three meeting moments:

moment $$$6$$$, during which both swimmers are in the point $$$6$$$;
moment $$$12$$$, during which both swimmers are in the point $$$6$$$;
and moment $$$18$$$, during which both swimmers are in the point $$$0$$$.

Informação

Provedor Codeforces

Código CF1845F