Problem:
Score : 400 points

Problem Statement

Your friend gave you a dequeue D as a birthday present.

D is a horizontal cylinder that contains a row of N jewels.

The values of the jewels are V_1, V_2, ..., V_N from left to right. There may be jewels with negative values.

In the beginning, you have no jewel in your hands.

You can perform at most K operations on D, chosen from the following, at most K times (possibly zero):

Operation A: Take out the leftmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.

Operation B: Take out the rightmost jewel contained in D and have it in your hand. You cannot do this operation when D is empty.

Operation C: Choose a jewel in your hands and insert it to the left end of D. You cannot do this operation when you have no jewel in your hand.

Operation D: Choose a jewel in your hands and insert it to the right end of D. You cannot do this operation when you have no jewel in your hand.

Find the maximum possible sum of the values of jewels in your hands after the operations.

Constraints

All values in input are integers.

1 \leq N \leq 50

1 \leq K \leq 100

-10^7 \leq V_i \leq 10^7

Input

Input is given from Standard Input in the following format:

N K
V_1 V_2 ... V_N

Output

Print the maximum possible sum of the values of jewels in your hands after the operations.

Sample Input 1

6 4
-10 8 2 1 2 6

Sample Output 1

14

After the following sequence of operations, you have two jewels of values 8 and 6 in your hands for a total of 14, which is the maximum result.

Do operation A. You take out the jewel of value -10 from the left end of D.

Do operation B. You take out the jewel of value 6 from the right end of D.

Do operation A. You take out the jewel of value 8 from the left end of D.

Do operation D. You insert the jewel of value -10 to the right end of D.

Sample Input 2

6 4
-6 -100 50 -2 -5 -3

Sample Output 2

44

Sample Input 3

6 3
-6 -100 50 -2 -5 -3

Sample Output 3

0

It is optimal to do no operation.

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

// -*- coding:utf-8-unix -*-

extern crate lazy_static;
extern crate num_bigint;
// 0.2.2
extern crate num_traits;

// 0.2.8
use num_bigint::BigInt;
use num_traits::{one, zero, Num, NumAssignOps, NumOps, One, Pow, Zero, abs};

// use proconio::derive_readable;
use proconio::fastout;
use proconio::input;
// use std::convert::TryInto;
use itertools::{assert_equal, concat, Itertools};
use lazy_static::lazy_static;
// use libm::*;
use std::cmp::*;
use std::collections::{BinaryHeap, HashMap, HashSet, VecDeque};
use std::io::*;
use std::mem::swap;
use std::ops::{BitAnd, Range, ShrAssign};
use std::str::FromStr;
use std::sync::Mutex;
use superslice::*;
use itertools_num::ItertoolsNum;

// ##########
// read
// ###########
pub fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.expect("failed to read char") as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().expect("failed to parse token")
}

// ##########
// chmin, chmax
// https://qiita.com/maguro_tuna/items/fab200fdc1efde1612e7
// ###########

#[allow(unused_macros)]
macro_rules! chmin {
    ($base:expr, $($cmps:expr),+ $(,)*) => {{
        let cmp_min = min!($($cmps),+);
        if $base > cmp_min {
            $base = cmp_min;
            true
        } else {
            false
        }
    }};
}

#[allow(unused_macros)]
macro_rules! chmax {
    ($base:expr, $($cmps:expr),+ $(,)*) => {{
        let cmp_max = max!($($cmps),+);
        if $base < cmp_max {
            $base = cmp_max;
            true
        } else {
            false
        }
    }};
}

#[allow(unused_macros)]
macro_rules! min {
    ($a:expr $(,)*) => {{
        $a
    }};
    ($a:expr, $b:expr $(,)*) => {{
        std::cmp::min($a, $b)
    }};
    ($a:expr, $($rest:expr),+ $(,)*) => {{
        std::cmp::min($a, min!($($rest),+))
    }};
}

#[allow(unused_macros)]
macro_rules! max {
    ($a:expr $(,)*) => {{
        $a
    }};
    ($a:expr, $b:expr $(,)*) => {{
        std::cmp::max($a, $b)
    }};
    ($a:expr, $($rest:expr),+ $(,)*) => {{
        std::cmp::max($a, max!($($rest),+))
    }};
}



// ##########
// modint
// https://qiita.com/drken/items/3b4fdf0a78e7a138cd9a
// ##########

#[allow(dead_code)]
fn modinv<T: Num + NumAssignOps + NumOps + Copy + PartialOrd>(a: T, m: T) -> T {
    let mut a = a;
    let mut b = m;
    let mut u: T = one();
    let mut v: T = zero();

    while b != zero() {
        let t = a / b;
        a -= t * b;
        swap(&mut a, &mut b);
        u -= t * v;
        swap(&mut u, &mut v);
    }
    u %= m;
    if u < zero() {
        u += m;
    }
    return u;
}

#[test]
fn modinv_test() {
    assert_eq!(1, modinv(1, 13));
    assert_eq!(2, modinv(7, 13));
    assert_eq!(3, modinv(9, 13));
    assert_eq!(4, modinv(10, 13));
    assert_eq!(5, modinv(8, 13));
}

// long long modpow(long long a, long long n, long long mod) {
// long long res = 1;
// while (n > 0) {
// if (n & 1) res = res * a % mod;
// a = a * a % mod;
// n >>= 1;
// }
// return res;
// }
#[allow(dead_code)]
fn modpow<T>(a: T, n: T, modulo: T) -> T
where
    T: Num + NumAssignOps + NumOps + Copy + PartialOrd + BitAnd + PartialEq + ShrAssign,
    <T as BitAnd>::Output: PartialEq + Num,
{
    let mut res = one();
    let mut a = a;
    let mut n = n;
    while n > zero() {
        if (n & one()) == one() {
            res = res * a % modulo;
        }
        a = a * a % modulo;
        n >>= one();
    }

    return res;
}

#[test]
fn modpow_test() {
    assert_eq!(4, modpow(2, 2, 13));
    assert_eq!(3, modpow(2, 4, 13));
}

// 前処理 com_init(): O(n)
// クエリ処理 COM(n, k): O(1)
// conv::com_init();
// conv::com(n,k);
mod comb {
    use super::*;
    lazy_static! {
        static ref FAC: Mutex<Vec<usize>> = Mutex::default();
        static ref FINV: Mutex<Vec<usize>> = Mutex::default();
        static ref INV: Mutex<Vec<usize>> = Mutex::default();
        static ref MODULO: Mutex<usize> = Mutex::default();
        // static ref MAXNCONV: Mutex<usize> = Mutex::default();
    }

    // // テーブルを作る前処理
    // void com_init() {
    // fac[0] = fac[1] = 1;
    // finv[0] = finv[1] = 1;
    // inv[1] = 1;
    // for (int i = 2; i < MAX; i++){
    // fac[i] = fac[i - 1] * i % MOD;
    // inv[i] = MOD - inv[MOD%i] * (MOD / i) % MOD;
    // finv[i] = finv[i - 1] * inv[i] % MOD;
    // }

    #[allow(dead_code)]
    fn com_init_with(modulo: usize, maxn_conv: usize) {
        let mut fac = FAC.lock().unwrap();
        let mut finv = FINV.lock().unwrap();
        let mut inv = INV.lock().unwrap();
        *fac = vec![0; maxn_conv];
        *finv = vec![0; maxn_conv];
        *inv = vec![0; maxn_conv];

        let mut g_modulo = MODULO.lock().unwrap();
        *g_modulo = modulo;

        fac[0] = 1;
        fac[1] = 1;
        finv[0] = 1;
        finv[1] = 1;
        inv[1] = 1;

        for i in 2..maxn_conv {
            fac[i] = fac[i - 1] * i % modulo;
            inv[i] = modulo - inv[modulo % i] * (modulo / i) % modulo;
            finv[i] = finv[i - 1] * inv[i] % modulo;
        }
    }

    #[allow(dead_code)]
    pub fn com_init() {
        com_init_with(MOD, MAXN_CONV);
    }

    // // 二項係数計算
    // long long COM(int n, int k){
    // if (n < k) return 0;
    // if (n < 0 || k < 0) return 0;
    // return fac[n] * (finv[k] * finv[n - k] % MOD) % MOD;
    // }
    #[allow(dead_code)]
    pub fn com(n: usize, k: usize) -> usize {
        let fac = FAC.lock().unwrap();
        let finv = FINV.lock().unwrap();
        // let mut inv = INV.lock().unwrap();
        let m = *MODULO.lock().unwrap();

        if n < k {
            return 0;
        }
        // if n < 0 || k < 0 {
        //     return 0;
        // }
        return fac[n] * (finv[k] * finv[n - k] % m) % m;
    }

    #[test]
    fn com_test() {
        com_init_with(13, 100);
        assert_eq!(12, com(12, 1));
        assert_eq!(66 % 13, com(12, 2));
        assert_eq!(220 % 13, com(12, 3));
        assert_eq!(495 % 13, com(12, 4));
        assert_eq!(792 % 13, com(12, 5));
        assert_eq!(924 % 13, com(12, 6));
        assert_eq!(com(12, 5), com(12, 7));
    }
}

// ##########
// union-find
// http://sntea.hatenablog.com/entry/2017/06/07/091246
// ##########

mod uf {
    // let mut uf = uf::UnionFind::new(10);
    // uf.unite; uf.same
    #[allow(dead_code)]
    #[derive(Debug)]
    pub struct UnionFind {
    par: Vec<i64>,
        rank: Vec<usize>,
    }

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

        #[allow(dead_code)]
        fn find(&mut self, x: usize) -> usize {
            if self.par[x] < 0 {
                x
            } else {
                let par = self.par[x];
                let res = self.find(par as usize);
                self.par[x] = res as i64;
                res
            }
        }

        #[allow(dead_code)]
        pub fn same(&mut self, a: usize, b: usize) -> bool {
            self.find(a) == self.find(b)
        }

        #[allow(dead_code)]
        pub fn unite(&mut self, a: usize, b: usize) {
            let apar = self.find(a);
            let bpar = self.find(b);
            if self.rank[apar] > self.rank[bpar] {
                self.par[apar] += self.par[bpar];
                self.par[bpar] = apar as i64;
            } else {
                self.par[bpar] += self.par[apar];
                self.par[apar] = bpar as i64;
                if self.rank[apar] == self.rank[bpar] {
                    self.rank[bpar] += 1;
                }
            }
        }

        #[allow(dead_code)]
        pub fn size(&mut self, x: usize) -> usize {
            let parent = self.find(x);
            //parentのparにサイズが負の状態で入る
            return (-self.par[parent]) as usize;
        }
    }

    #[test]
    fn union_find_test() {
        let mut uf = UnionFind::new(10);

        for i in 0..10 {
            for j in 0..10 {
                assert_eq!(i == j, uf.same(i, j));
            }
        }

        uf.unite(0, 1);
        assert_eq!(true, uf.same(0, 1));
        //false
        assert_eq!(false, uf.same(0, 9));
        assert_eq!(false, uf.same(1, 9));
        assert_eq!(false, uf.same(2, 9));

        assert_eq!(2, uf.size(0));
        assert_eq!(2, uf.size(1));
        //1
        assert_eq!(1, uf.size(2));
        assert_eq!(1, uf.size(8));
        assert_eq!(1, uf.size(9));

        uf.unite(8, 9);
        assert_eq!(true, uf.same(0, 1));
        assert_eq!(true, uf.same(8, 9));
        //false
        assert_eq!(false, uf.same(0, 9));
        assert_eq!(false, uf.same(1, 9));
        assert_eq!(false, uf.same(2, 9));

        assert_eq!(2, uf.size(0));
        assert_eq!(2, uf.size(1));
        assert_eq!(2, uf.size(8));
        assert_eq!(2, uf.size(9));
        //1
        assert_eq!(1, uf.size(2));

        uf.unite(1, 9);
        assert_eq!(true, uf.same(0, 1));
        assert_eq!(true, uf.same(8, 9));
        assert_eq!(true, uf.same(0, 8));
        assert_eq!(true, uf.same(0, 9));
        assert_eq!(true, uf.same(1, 8));
        assert_eq!(true, uf.same(1, 9));
        //false
        assert_eq!(false, uf.same(2, 9));

        assert_eq!(4, uf.size(0));
        assert_eq!(4, uf.size(1));
        assert_eq!(4, uf.size(8));
        assert_eq!(4, uf.size(9));
        //1
        assert_eq!(1, uf.size(2));
    }
}

// ###########
// seg_tree
// ant_book
// ###########
mod seg_tree {
    #[derive(Debug)]
    struct SegTree<T: Clone> {
        n: usize,
        dat: Vec<Option<T>>,
    }

    impl<T: Clone + std::fmt::Debug> SegTree<T> {
        #[allow(dead_code)]
        pub fn new(size: usize) -> SegTree<T> {
            let mut size_pow2 = 1;
            while size_pow2 < size {
                size_pow2 *= 2;
            }

            let dat: Vec<Option<T>> = vec![None; 2 * size_pow2 - 1];
            SegTree { n: size_pow2, dat }
        }

        #[allow(dead_code)]
        pub fn update<F: Fn(&Option<T>, &Option<T>) -> Option<T>>(
            &mut self,
            k: usize,
            a: T,
            update: F,
        ) {
            let mut k = k;
            k += self.n - 1;
            self.dat[k] = Some(a);
            while k > 0 {
                k = (k - 1) / 2;
                self.dat[k] = update(&self.dat[k * 2 + 1], &self.dat[k * 2 + 2]);
            }
        }

        #[allow(dead_code)]
        fn query_inner<F: Fn(&Option<T>, &Option<T>) -> Option<T>>(
            &self,
            selection_query: &F,
            a: usize,
            b: usize,
            k: usize,
            l: usize,
            r: usize,
        ) -> Option<T> {
            if r <= a || b <= l {
                // eprintln!("{}, {}, {}, {}, {:?}", a, b, l, r, "none");
                return None;
            }

            return if a <= l && r <= b {
                // eprintln!("{}, {}, {}, {}, {:?}", a, b, r, l, self.dat[k]);
                self.dat[k].clone()
            } else {
                let vl = self.query_inner(selection_query, a, b, k * 2 + 1, l, (l + r) / 2);
                let vr = self.query_inner(selection_query, a, b, k * 2 + 2, (l + r) / 2, r);

                selection_query(&vl, &vr)
            };
        }

        #[allow(dead_code)]
        pub fn query<F: Fn(&Option<T>, &Option<T>) -> Option<T>>(
            &self,
            selection_query: &F,
            a: usize,
            b: usize,
        ) -> Option<T> {
            return self.query_inner(selection_query, a, b, 0, 0, self.n);
        }
    }

    #[test]
    fn test_segtree_rmq() {
        let mut t: SegTree<usize> = SegTree::new(5);

        let cmp_f = |lhs: &Option<usize>, rhs: &Option<usize>| {
            if lhs.is_none() {
                return rhs.clone();
            }

            if rhs.is_none() {
                return lhs.clone();
            }

            return if lhs.unwrap() <= rhs.unwrap() {
                lhs.clone()
            } else {
                rhs.clone()
            };
        };
        // 1, 3, 2, 5, 1
        t.update(0, 1, cmp_f);
        t.update(1, 3, cmp_f);
        t.update(2, 2, cmp_f);
        t.update(3, 5, cmp_f);
        t.update(4, 1, cmp_f);
        // println!("{:?}", t);

        assert_eq!(1, t.query(&cmp_f, 0, 1).unwrap());
        assert_eq!(3, t.query(&cmp_f, 1, 2).unwrap());
        assert_eq!(2, t.query(&cmp_f, 2, 3).unwrap());
        assert_eq!(5, t.query(&cmp_f, 3, 4).unwrap());
        assert_eq!(1, t.query(&cmp_f, 4, 5).unwrap());

        // len2
        assert_eq!(1, t.query(&cmp_f, 0, 2).unwrap());
        assert_eq!(2, t.query(&cmp_f, 1, 3).unwrap());
        assert_eq!(2, t.query(&cmp_f, 2, 4).unwrap());
        assert_eq!(1, t.query(&cmp_f, 3, 5).unwrap());

        // len3
        assert_eq!(1, t.query(&cmp_f, 0, 3).unwrap());
        assert_eq!(2, t.query(&cmp_f, 1, 4).unwrap());
        assert_eq!(1, t.query(&cmp_f, 2, 5).unwrap());

        // len4
        assert_eq!(1, t.query(&cmp_f, 0, 4).unwrap());
        assert_eq!(1, t.query(&cmp_f, 1, 5).unwrap());

        // len5
        assert_eq!(1, t.query(&cmp_f, 0, 6).unwrap());
    }
}

// ##############
// rolling hash
// ###############
mod rolling_hash {
    use super::*;
    use ascii::{AsciiStr, AsciiString};
    use num_traits::AsPrimitive;

    fn contains_with(base: u64, a: &AsciiStr, b: &AsciiStr) -> bool {
        let (al, bl) = (a.len(), b.len());
        if al > bl {
            return false;
        }

        let mut t: u64 = 1;
        for _ in 0..al {
            t = t.wrapping_mul(base);
        }

        let (mut ah, mut bh): (u64, u64) = (0, 0);

        for i in 0..al {
            ah = ah.wrapping_mul(base) + a[i].as_byte() as u64;
        }
        for i in 0..al {
            bh = bh.wrapping_mul(base) + b[i].as_byte() as u64;
        }

        // eprintln!("{}, {}", ah, bh);

        for i in 0..=bl - al {
            if ah == bh {
                return true;
            }
            if i + al < bl {
                let mut add: i64 = b[i + al].as_byte().as_();
                add -= ((b[i].as_byte() as u64).wrapping_mul(t)) as i64;
                bh = (bh.wrapping_mul(base) as i64).wrapping_add(add) as u64;
            }
        }

        return false;
    }

    #[allow(dead_code)]
    pub fn contains(a: &AsciiStr, b: &AsciiStr) -> bool {
        return contains_with(BASE_ROLLING_HASH, a, b);
    }

    #[test]
    fn contains_test() {
        const base: u64 = 1000_000_007;
        assert_eq!(
            false,
            contains_with(
                base,
                &AsciiString::from_str("abc").unwrap(),
                &AsciiString::from_str("a").unwrap()
            )
        );

        assert_eq!(
            true,
            contains_with(
                base,
                &AsciiString::from_str("abc").unwrap(),
                &AsciiString::from_str("aaabca").unwrap()
            )
        );

        assert_eq!(
            true,
            contains_with(
                base,
                &AsciiString::from_str("aaaaaa").unwrap(),
                &AsciiString::from_str("aaaaaa").unwrap()
            )
        );

        assert_eq!(
            false,
            contains_with(
                base,
                &AsciiString::from_str("abc").unwrap(),
                &AsciiString::from_str("aacbaa").unwrap()
            )
        )
    }

    fn overlap_last_and_head_with(base: u64, a: &AsciiStr, b: &AsciiStr) -> usize {
        let (al, bl) = (a.len(), b.len());

        let mut ans = 0;
        let (mut ah, mut bh, mut t): (u64, u64, u64) = (0, 0, 1);
        for i in 1..=min(al, bl) {
            ah = ah.wrapping_add((a[al - i].as_byte() as u64).wrapping_mul(t));
            bh = bh
                .wrapping_mul(base)
                .wrapping_add(b[i - 1].as_byte() as u64);

            if ah == bh {
                ans = i;
            }
            t = t.wrapping_mul(base);
        }

        return ans;
    }

    fn overlap_last_and_head(a: &AsciiStr, b: &AsciiStr) -> usize {
        return overlap_last_and_head_with(BASE_ROLLING_HASH, a, b);
    }

    #[test]
    fn overlap_test() {
        const base: u64 = 1000_000_007;
        assert_eq!(
            0,
            overlap_last_and_head_with(
                base,
                &AsciiString::from_str("abc").unwrap(),
                &AsciiString::from_str("a").unwrap()
            )
        );

        assert_eq!(
            2,
            overlap_last_and_head_with(
                base,
                &AsciiString::from_str("abc").unwrap(),
                &AsciiString::from_str("bca").unwrap()
            )
        );

        assert_eq!(
            5,
            overlap_last_and_head_with(
                base,
                &AsciiString::from_str("hogefoobar").unwrap(),
                &AsciiString::from_str("oobarhoge").unwrap()
            )
        );
    }
}

// ##########
// lazy_static!
// ##########
// lazy_static! {
//     static ref H: Mutex<Vec<i32>> = Mutex::default();
//     static ref W: Mutex<Vec<i32>> = Mutex::default();
// }
// let mut values = VALUES.lock().unwrap();
// values.extend_from_slice(&[1, 2, 3, 4]);
// assert_eq!(&*values, &[1, 2, 3, 4]);

// let mut values = VALUES.lock().unwrap();
// values.extend_from_slice(&[1, 2, 3, 4]);

// MOD, Combination関連に使う定数
#[allow(dead_code)]
const BASE_ROLLING_HASH: u64 = 1158187049;
#[allow(dead_code)]
const MOD: usize = 1000000007;
#[allow(dead_code)]
const MAXN_CONV: usize = 510000;

// abc128-D
// #[fastout]
fn main() {
    input![n:usize, k: usize, v: [i64; n]];
    //new type

    // let l_cumsum: Vec<i64> = v.iter().cumsum().collect_vec();

    let mut ans = 0;
    for l in 0..=k{
        for r in 0..=k{

            let mut new_ans = 0;

            if (l + r) > k || (l + r) > n{
                continue;
            }

            let mut h = BinaryHeap::new();
            for i in 0..l{
                new_ans += v[i];
                if v[i] < 0 {
                    h.push(-v[i]);
                }
            }

            for i in 0..r{
                let diff = v[v.len() - 1 - i];
                new_ans += diff;
                if diff < 0 {
                    h.push(-diff);
                }
            }

            for _ in 0..(k-l-r){
                if let Some(v) = h.pop(){
                    new_ans += v
                }else{
                    break;
                }
            }

            // eprintln!("l: {}, r: {}, ans: {}", l, r, new_ans);

            chmax!(ans, new_ans);
        }
    }

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