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 Kotlin (1.3.71) (judge verdict: Time Limit Exceeded)

import java.io.*
import java.util.*
import java.lang.*
import java.lang.Math.*


fun solve() {
    val (N,K) = rd.readListInt()
    val A = rd.readListLong()

    val ms = mutableSetOf<Long>()
    for (i in 0 until N-1) {
        var sum = A[i]
        ms.add(sum)
        for (j in i+1 until N) {
            sum += A[j]
            ms.add(sum)
        }
    }

    val bt = ms.toList()
    val M = bt.size
    val B = Array(M){""}
    for (i in 0 until M) {
        B[i] = bt[i].toString(2).padStart(64, '0')
    }

    val RB = Array(64){IntArray(M)}
    for (i in 0 until M) {
        for (d in 0 until 64) {
            RB[d][i] = B[i][d] - '0'
        }
    }

    val take = RB.map{it.count{it>0} >= K}

    val range = RB.map{
        it.mapIndexed{i,v ->
            if (v>0) i else -1}
        .filter { it>=0 }.toMutableSet()
    }

    val res = Array(64){mutableSetOf<Int>()}
    var pd = -1
    for (d in 0 until 64) {
        if (take[d]) {
            if (pd < 0) {
                res[d] = range[d]
                pd = d
            } else {
                val prod = res[pd].intersect(range[d])
                if (prod.size >= K) {
                    res[d].addAll(prod)
                    pd = d
                }
            }
        }
    }

    var ans = 0L
    for (d in 0 until 64) {
        if (res[d].size >=K) {
            ans += 1L shl (63-d)
        }
    }

    println(ans)

    return
}


//val rd = debug.Reader("etc", "dwacon5th-prelims", "b", "sample-1")
val rd = Reader()


fun main(args: Array<String>) {
    try {
        var exception: Throwable? = null
        val sth = Thread(null, SolveThread(), "solveThread", 128 * 1024 * 1024)
        sth.setUncaughtExceptionHandler { _, e -> exception = e }
        sth.start(); sth.join()
        if (exception != null) throw exception!! }
    catch (e: Exception) { throw e }
    finally { rd.close() }
}
const val MOD = 1_000_000_007L; const val INF = Int.MAX_VALUE/2; const val LINF = Long.MAX_VALUE/2
const val EPS = 1e-9; fun DBLEQ(a: Double, b: Double): Boolean {return abs(a - b) < EPS}
data class IntPair(val first: Int, val second: Int); data class LongPair(val first: Long, val second: Long)
data class To(val idx: Int, val cost: Long)
typealias ALI = ArrayList<Int>
fun <T>edge(V: Int): Array<ArrayList<T>> { return Array(V){ArrayList<T>()} }
val errPrintln: (Any) -> Unit = { msg -> System.err.println(msg) }
fun rAssert(exp: Boolean, lazyMsg: () -> String = {""}) {
    if (!exp) { val msg = lazyMsg(); errPrintln("EAssert Error: $msg"); throw Error(msg) }
}
fun tAssert(exp: Boolean, lazyMsg: () -> String = {""}) {
    if (!exp) { val msg = lazyMsg(); errPrintln("TAssert Error: $msg"); Thread.sleep(3000) }
}
class SolveThread : Runnable {
    override fun run() { solve() }
}
class Reader {
    private val br = BufferedReader(InputStreamReader(System.`in`))
    val readString: () -> String = { br.readLine() }
    val readInt: () -> Int = { br.readLine().toInt() }
    val readLong: () -> Long = { br.readLine().toLong() }
    val readDouble: () -> Double = { br.readLine().toDouble() }
    val readListInt: () -> List<Int> = { br.readLine().split(' ').map { it.toInt() } }
    val readListLong: () -> List<Long> = { br.readLine().split(' ').map { it.toLong() } }
    val readIntArray: () -> IntArray = { readListInt().toIntArray() }
    val readLongArray: () -> LongArray = { readListLong().toLongArray() }
    val readListStringCol: (Int) -> List<String> = { N -> (0 until N).map { readString() } }
    val readListIntCol: (Int) -> List<Int> = { N -> (0 until N).map { readInt() } }
    val readListIntCols: (Int) -> List<List<Int>> = { N -> (0 until N).map { readListInt() } }
    val readListLongCol: (Int) -> List<Long> = { N -> (0 until N).map { readLong() } }
    val readListLongCols: (Int) -> List<List<Long>> = { N -> (0 until N).map { readListLong() } }
    val close: () -> Unit = { br.close() }
    protected fun finalize() { close() }
}