Problem:
Score : 600 points

Problem Statement

Snuke has a blackboard and a set S consisting of N integers.
The i-th element in S is S_i.

He wrote an integer X on the blackboard, then performed the following operation N times:

Choose one element from S and remove it.

Let x be the number written on the blackboard now, and y be the integer removed from S. Replace the number on the blackboard with x \bmod {y}.

There are N! possible orders in which the elements are removed from S.
For each of them, find the number that would be written on the blackboard after the N operations, and compute the sum of all those N! numbers modulo 10^{9}+7.

Constraints

All values in input are integers.

1 \leq N \leq 200

1 \leq S_i, X \leq 10^{5}

S_i are pairwise distinct.

Input

Input is given from Standard Input in the following format:

N X
S_1 S_2 \ldots S_{N}

Output

Print the answer.

Sample Input 1

2 19
3 7

Sample Output 1

3

There are two possible orders in which we remove the numbers from S.

If we remove 3 and 7 in this order, the number on the blackboard changes as follows: 19 \rightarrow 1 \rightarrow 1.

If we remove 7 and 3 in this order, the number on the blackboard changes as follows: 19 \rightarrow 5 \rightarrow 2.

The output should be the sum of these: 3.

Sample Input 2

5 82
22 11 6 5 13

Sample Output 2

288

Sample Input 3

10 100000
50000 50001 50002 50003 50004 50005 50006 50007 50008 50009

Sample Output 3

279669259

Be sure to compute the sum modulo 10^{9}+7.

Solution in Rust (1.42.0) (judge verdict: Accepted)

use proconio::*;

fn run() {
    input! {
        n: usize,
        w: usize,
        mut s: [usize; n],
    }
    s.sort();
    const MOD: usize = 1_000_000_007;
    let mut dp: Vec<usize> = (0..=w).collect();
    for (i, a) in s.into_iter().enumerate() {
        if a <= w {
            let (x, y) = dp.split_at_mut(a);
            for (y, x) in y.iter_mut().zip(x.iter().cycle()) {
                *y = (*y * i + *x) % MOD;
            }
            x.iter_mut().for_each(|x| *x = *x * (i + 1) % MOD);
        } else {
            dp.iter_mut().for_each(|p| *p = (i + 1) * *p % MOD);
        }
    }
    let ans = dp[w];
    println!("{}", ans);
}

fn main() {
    run();
}