Problem:
Score : 400 points

Problem Statement

One day, Niwango-kun, an employee of Dwango Co., Ltd., found an integer sequence (a_1, ..., a_N) of length N.
He is interested in properties of the sequence a.

For a nonempty contiguous subsequence a_l, ..., a_r (1 \leq l \leq r \leq N) of the sequence a, its beauty is defined as a_l + ... + a_r. Niwango-kun wants to know the maximum possible value of the bitwise AND of the beauties of K nonempty contiguous subsequences among all N(N+1)/2 nonempty contiguous subsequences. (Subsequences may share elements.)

Find the maximum possible value for him.

Constraints

2 \leq N \leq 1000

1 \leq a_i \leq 10^9

1 \leq K \leq N(N+1)/2

All numbers given in input are integers

Input

Input is given from Standard Input in the following format:

N K
a_1 a_2 ... a_N

Output

Print the answer.

Sample Input 1

4 2
2 5 2 5

Sample Output 1

12

There are 10 nonempty contiguous subsequences of a. Let us enumerate them:

contiguous subsequences starting from the first element: \{2\}, \{2, 5\}, \{2, 5, 2\}, \{2, 5, 2, 5\}

contiguous subsequences starting from the second element: \{5\}, \{5, 2\}, \{5, 2, 5\}

contiguous subsequences starting from the third element: \{2\}, \{2, 5\}

contiguous subsequences starting from the fourth element: \{5\}

(Note that even if the elements of subsequences are equal, subsequences that have different starting indices are considered to be different.)

The maximum possible bitwise AND of the beauties of two different contiguous subsequences is 12.
This can be achieved by choosing \{5, 2, 5\} (with beauty 12) and \{2, 5, 2, 5\} (with beauty 14).

Sample Input 2

8 4
9 1 8 2 7 5 6 4

Sample Output 2

32

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

#[allow(unused_macros, dead_code)]
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        let mut next = || { iter.next().unwrap() };
        input_inner!{next, $($r)*}
    };
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

#[allow(unused_macros, dead_code)]
macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };

    ($next:expr, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

#[allow(unused_macros, dead_code)]
macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, bytes) => {
        read_value!($next, String).into_bytes()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

#[allow(dead_code)]
struct UnionFind {
    parent: Vec<usize>,
    rank: Vec<usize>,
    size: Vec<usize>,
}

#[allow(dead_code)]
impl UnionFind {
    fn new(n: usize) -> UnionFind {
        let mut p = vec![0; n];
        for i in 0..n {
            p[i] = i;
        }
        return UnionFind {
            parent: p,
            rank: vec![0; n],
            size: vec![1; n],
        };
    }

    fn find(&mut self, x: usize) -> usize {
        if x == self.parent[x] {
            x
        } else {
            let p = self.parent[x];
            let pr = self.find(p);
            self.parent[x] = pr;
            pr
        }
    }

    fn same(&mut self, a: usize, b: usize) -> bool {
        self.find(a) == self.find(b)
    }

    fn unite(&mut self, a: usize, b: usize) {
        let a_root = self.find(a);
        let b_root = self.find(b);
        if a_root == b_root {
            return;
        }
        if self.rank[a_root] > self.rank[b_root] {
            self.parent[b_root] = a_root;
            self.size[a_root] += self.size[b_root];
        } else {
            self.parent[a_root] = b_root;
            self.size[b_root] += self.size[a_root];
            if self.rank[a_root] == self.rank[b_root] {
                self.rank[b_root] += 1;
            }
        }
    }

    fn get_size(&mut self, x: usize) -> usize {
        let root = self.find(x);
        self.size[root]
    }
}

const MOD_P: usize = 1000000007;

#[allow(dead_code)]
// fact(n) = n! mod p
fn fact(n: usize) -> usize {
    let mut acc = 1;
    for i in 1..n + 1 {
        acc = acc * i % MOD_P;
    }
    acc
}

#[allow(dead_code)]
fn mod_pow(b: usize, mut e: usize) -> usize {
    let mut base = b;
    let mut acc = 1;

    while e > 1 {
        if e % 2 == 1 {
            acc = acc * base % MOD_P;
        }
        e /= 2;
        base = base * base % MOD_P;
    }

    if e == 1 {
        acc = acc * base % MOD_P;
    }

    acc
}

#[allow(dead_code)]
fn comb(n: usize, r: usize) -> usize {
    // nCr = n! / (r! (n-r)!) = n! (r!)^(p-2) ((n-r)!)^(p-2)
    let x = ((n - r + 1)..(n + 1)).fold(1, |p, x| p * x % MOD_P);
    let y = mod_pow(fact(r), MOD_P - 2);
    x * y % MOD_P
}

#[derive(Clone, Copy, Debug)]
struct GF(usize);

impl std::ops::Add for GF {
    type Output = GF;
    fn add(self, rhs: GF) -> Self::Output {
        let mut d = self.0 + rhs.0;
        if d >= MOD_P {
            d -= MOD_P;
        }
        GF(d)
    }
}

impl std::ops::AddAssign for GF {
    fn add_assign(&mut self, rhs: GF) {
        *self = *self + rhs;
    }
}

impl std::ops::Sub for GF {
    type Output = GF;
    fn sub(self, rhs: GF) -> Self::Output {
        let mut d = self.0 + MOD_P - rhs.0;
        if d >= MOD_P {
            d -= MOD_P;
        }
        GF(d)
    }
}

impl std::ops::SubAssign for GF {
    fn sub_assign(&mut self, rhs: GF) {
        *self = *self - rhs;
    }
}

impl std::ops::Mul for GF {
    type Output = GF;
    fn mul(self, rhs: GF) -> Self::Output {
        let mut d = self.0 * rhs.0;
        d %= MOD_P;
        GF(d)
    }
}

impl std::ops::MulAssign for GF {
    fn mul_assign(&mut self, rhs: GF) {
        *self = *self * rhs;
    }
}

impl std::fmt::Display for GF {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

#[allow(dead_code)]
impl GF {
    pub fn new(n: usize) -> GF {
        GF(n % MOD_P)
    }
    pub fn zero() -> GF {
        GF(0)
    }
    pub fn one() -> GF {
        GF(1)
    }
    pub fn pow(self, mut e: usize) -> GF {
        let mut acc = GF::one();
        let mut b = self;
        while e > 1 {
            if e % 2 == 1 {
                acc *= b;
            }
            b *= b;
            e /= 2;
        }
        if e == 1 {
            acc *= b;
        }
        acc
    }
    pub fn fact(self) -> GF {
        let mut acc = GF::one();
        for i in 1..=self.0 {
            acc *= GF::new(i);
        }
        acc
    }
    pub fn inv(self) -> GF {
        self.pow(MOD_P - 2)
    }
    pub fn comb(n: GF, r: GF) -> GF {
        // nCr = n! / (r! (n-r)!) = n! (r!)^(p-2) ((n-r)!)^(p-2)
        let x = ((n.0 - r.0 + 1)..=n.0).fold(GF::one(), |p, x| p * GF(x));
        let y = r.fact().inv();
        x * y
    }
}

#[allow(dead_code)]
#[derive(Debug)]
struct MemComb {
    inv: Vec<GF>,
    fact: Vec<GF>,
    factinv: Vec<GF>,
}

#[allow(dead_code)]
impl MemComb {
    pub fn new(n: usize) -> MemComb {
        let mut inv = vec![GF::one(); n + 1];
        let mut fact = vec![GF::one(); n + 1];
        let mut factinv = vec![GF::one(); n + 1];

        for i in 2..=n {
            fact[i] = fact[i - 1] * GF(i);
        }

        factinv[n] = fact[n].inv();
        for i in (1..n).rev() {
            factinv[i] = factinv[i + 1] * GF(i + 1); // 1/n! = 1/(n+1)! * (n+1)
            inv[i] = factinv[i] * fact[i - 1]; // 1/n = 1/n! * (n-1)!
        }
        inv[n] = factinv[n] * fact[n - 1];

        MemComb { inv, fact, factinv }
    }
    pub fn comb(&self, n: usize, r: usize) -> GF {
        if n >= r {
            self.fact[n] * self.factinv[r] * self.factinv[n - r]
        } else {
            GF::zero()
        }
    }
}

#[allow(dead_code)]
fn gcd(a: usize, b: usize) -> usize {
    if b > a {
        gcd(b, a)
    } else if a % b == 0 {
        b
    } else {
        gcd(b, a % b)
    }
}

#[allow(dead_code)]
fn getline() -> String {
    let mut __ret = String::new();
    std::io::stdin().read_line(&mut __ret).ok();
    return __ret;
}

#[allow(unused_imports)]
use std::cmp::{max, min};

fn main() {
    input! {
        n: usize,
        k: usize,
        a: [usize; n],
    }

    let mut v = Vec::new();
    for i in 0..n {
        for j in i..n {
            let x = (i..=j).fold(0, |acc, x| acc + a[x]);
            v.push(x);
        }
    }
    // eprintln!("{:?}", v);
    let maximum = v.iter().fold(0, |acc, &x| max(acc, x));
    let mut bits = 0;
    while (1 << bits) <= maximum {
        bits += 1;
    }

    // eprintln!("{}({:0b})\t{}", maximum, maximum, bits);
    let mut ans = 0;
    'outer: for i in (0..bits).rev() {
        let ans1 = ans | (1 << i);
        let mut c = 0;
        for &x in &v {
            if (x & ans1) == ans1 {
                c += 1;
            }
            if c >= k {
                ans = ans1;
                continue 'outer;
            }
        }
    }

    println!("{}", ans);
}