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: Wrong Answer)

fun main() = dwacon5thprelimsB()

@OptIn(ExperimentalStdlibApi::class)
fun dwacon5thprelimsB() = IO_dwacon5thprelimsB().exec {
    val (n, k) = readInt() to readInt()
    val aList = LongArray(n) { readLong() }

    val subArraySums = mutableListOf<Long>()

    for (left in 0 until n) {
        var sum = 0L
        for (right in left until n) {
            sum += aList[right]
            subArraySums.add(sum)
        }
    }

    fun dfs(bitDigit: Int, subList: List<Long>, foundMoreThanK: Boolean): List<Long> {
        if (bitDigit == 0) return if (foundMoreThanK) subList else List(k) { 0L }

        val filtered = subList.filter { (it shr bitDigit.dec()) % 2 == 1L }
        return when {
            filtered.size == k -> filtered
            filtered.size > k -> dfs(bitDigit.dec(), filtered, true)
            else -> if (foundMoreThanK) subList else dfs(bitDigit.dec(), subList, false)
        }
    }

    val answer = dfs(60, subArraySums, false).take(k).reduce(Long::and)

    println(answer)
}

// region kokokara template dayo (^o^)
@Suppress("unused", "ClassName", "SpellCheckingInspection", "ConvertToStringTemplate", "MemberVisibilityCanBePrivate")
private class IO_dwacon5thprelimsB(private val separator: String = System.lineSeparator()) {
    private val input = System.`in`
    private val buffer = ByteArray(1024)
    private var pointer = 0
    private var bufferLength = 0
    private val sb = StringBuilder()
    private fun Byte.isPrintable() = this in 33..126
    private fun Byte.isNumeric() = this in '0'.toByte()..'9'.toByte()
    private fun Byte.toNumVal() = if (this.isNumeric()) this - '0'.toByte() else error("$this is not numeric")

    private fun hasNextByte(): Boolean {
        return if (pointer < bufferLength) true else {
            pointer = 0
            bufferLength = input.read(buffer)
            bufferLength > 0
        }
    }

    private fun readByte(): Byte = if (hasNextByte()) buffer[pointer++] else -1
    private fun skipUnprintable() = run { while (hasNextByte() && !buffer[pointer].isPrintable()) pointer++ }
    private fun hasNext(): Boolean = run { skipUnprintable() }.run { hasNextByte() }
    private fun hasNextOrError() = if (!hasNext()) error("has no next element.") else Unit

    fun readString(): String {
        hasNextOrError()
        val sb = StringBuilder()
        var b = readByte()
        while (b.isPrintable()) {
            sb.appendCodePoint(b.toInt())
            b = readByte()
        }
        return sb.toString()
    }

    fun readLong(): Long {
        hasNextOrError()
        var n = 0L
        var negative = false
        var b = readByte()
        if (b == '-'.toByte()) {
            negative = true
            b = readByte()
        }
        if (!b.isNumeric()) error("$b is not numeric.")
        while (true) {
            when {
                b.isNumeric() -> n = n * 10 + b.toNumVal()
                b.toInt() == -1 || !b.isPrintable() -> return if (negative) -n else n
                else -> error("failed to parse. [n=$n, b=$b]")
            }
            b = readByte()
        }
    }

    fun readInt() = readLong()
        .let { if (it in Int.MIN_VALUE..Int.MAX_VALUE) it.toInt() else error("$it is not in range of Int.") }

    fun readDouble(): Double {
        var n = 0.0
        var div = 1.0
        var negative = false
        var b = readByte()
        if (b == '-'.toByte()) {
            negative = true
            b = readByte()
        }
        do n = n * 10 + b.toNumVal()
        while (run { b = readByte() }.run { b.isNumeric() })
        if (b == '.'.toByte()) {
            while (run { b = readByte() }.run { b.isNumeric() })
                n += b.toNumVal() / (run { div *= 10 }.run { div })
        }
        return if (negative) -n else n
    }

    fun println(): Unit = run { sb.append(separator) }
    fun print(any: Any): Unit = run { sb.append(any) }
    fun println(any: Any): Unit = run { sb.append(any.toString() + separator) }
    fun flush() = run { kotlin.io.println(sb); sb.clear() }

    fun exec(action: IO_dwacon5thprelimsB.() -> Unit) {
        var t: Throwable? = null
        Thread(null, { action() }, "solve", 128 * 1024 * 1024)
            .apply { setUncaughtExceptionHandler { _, t1 -> t = t1 } }
            .apply { start() }.join()
        t?.let { throw it }
        kotlin.io.print(sb)
    }

    fun readLine(): Nothing = error("readLine is disabled.")
}
// endregion kokomade template dayo (^o^)