Problem:
Score : 500 points

Problem Statement

We have a sequence of N integers: A_1, A_2, \cdots, A_N.

You can perform the following operation between 0 and K times (inclusive):

Choose two integers i and j such that i \neq j, each between 1 and N (inclusive). Add 1 to A_i and -1 to A_j, possibly producing a negative element.

Compute the maximum possible positive integer that divides every element of A after the operations. Here a positive integer x divides an integer y if and only if there exists an integer z such that y = xz.

Constraints

2 \leq N \leq 500

1 \leq A_i \leq 10^6

0 \leq K \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
A_1 A_2 \cdots A_{N-1} A_{N}

Output

Print the maximum possible positive integer that divides every element of A after the operations.

Sample Input 1

2 3
8 20

Sample Output 1

7

7 will divide every element of A if, for example, we perform the following operation:

Choose i = 2, j = 1. A becomes (7, 21).

We cannot reach the situation where 8 or greater integer divides every element of A.

Sample Input 2

2 10
3 5

Sample Output 2

8

Consider performing the following five operations:

Choose i = 2, j = 1. A becomes (2, 6).

Choose i = 2, j = 1. A becomes (1, 7).

Choose i = 2, j = 1. A becomes (0, 8).

Choose i = 2, j = 1. A becomes (-1, 9).

Choose i = 1, j = 2. A becomes (0, 8).

Then, 0 = 8 \times 0 and 8 = 8 \times 1, so 8 divides every element of A. We cannot reach the situation where 9 or greater integer divides every element of A.

Sample Input 3

4 5
10 1 2 22

Sample Output 3

7

Sample Input 4

8 7
1 7 5 6 8 2 6 5

Sample Output 4

5

Solution in Rust (1.42.0) (judge verdict: Runtime Error)

#![allow(unused_imports)]
use {
   proconio::{fastout, input, marker::*},
   std::cmp::*,
   std::collections::*,
};

const INF: i64 = 1001001001001001001;

#[fastout]
fn main() {
   input! {
      n: usize,
      m: usize,
      p: i64,
      abc: [(Usize1, Usize1, i64); m]
   }
   let mut to = vec![vec![]; n];
   for (a, b, c) in abc {
      to[a].push((b, c));
   }

   let mut d = vec![-INF; n];
   d[0] = 0;
   for _ in 0..n {
      for v in 0..n {
         if d[v] == -INF {
            continue;
         }
         for &(u, c) in to[v].iter() {
            d[u] = max(d[u], d[v] + c - p);
         }
      }
   }
   let ans = max(0, d[n - 1]);
   for _ in 0..n {
      for v in 0..n {
         if d[v] == -INF {
            continue;
         }
         for &(u, c) in to[v].iter() {
            d[u] = max(d[u], d[v] + c - p);
         }
      }
   }
   if d[n - 1] == INF || ans < d[n - 1] {
      println!("{}", -1);
   } else {
      println!("{}", ans);
   }
}