Problem:
Score : 500 points

Problem Statement

Given are N integers A_1,\ldots,A_N.

We will choose exactly K of these elements. Find the maximum possible product of the chosen elements.

Then, print the maximum product modulo (10^9+7), using an integer between 0 and 10^9+6 (inclusive).

Constraints

1 \leq K \leq N \leq 2\times 10^5

|A_i| \leq 10^9

Input

Input is given from Standard Input in the following format:

N K
A_1 \ldots A_N

Output

Print the maximum product modulo (10^9+7), using an integer between 0 and 10^9+6 (inclusive).

Sample Input 1

4 2
1 2 -3 -4

Sample Output 1

12

The possible products of the two chosen elements are 2, -3, -4, -6, -8, and 12, so the maximum product is 12.

Sample Input 2

4 3
-1 -2 -3 -4

Sample Output 2

1000000001

The possible products of the three chosen elements are -24, -12, -8, and -6, so the maximum product is -6.

We print this value modulo (10^9+7), that is, 1000000001.

Sample Input 3

2 1
-1 1000000000

Sample Output 3

1000000000

The possible products of the one chosen element are -1 and 1000000000, so the maximum product is 1000000000.

Sample Input 4

10 10
1000000000 100000000 10000000 1000000 100000 10000 1000 100 10 1

Sample Output 4

999983200

Be sure to print the product modulo (10^9+7).

Solution in Java (OpenJDK 11.0.6) (judge verdict: Wrong Answer)

import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Collection;
import java.util.EnumSet;
import java.util.HashMap;
import java.util.NoSuchElementException;
import java.util.function.IntUnaryOperator;
import java.util.function.LongBinaryOperator;
import java.util.function.LongUnaryOperator;
import static java.lang.Math.abs;



public class Main {
    public static void main(String[] args) {
        StringBuilder out = new StringBuilder();
        solve(out);
        PrintWriter pw = new PrintWriter(System.out);
        pw.println(out);
        pw.flush();
        pw.close();
    }

    public static void solve(StringBuilder out) {
        ModuloArithmetic ma = new ModuloArithmetic(Const.MOD7);
        int n = In.ni();
        int k = In.ni();
        long[] a = In.nl(n);
        long ans = 1;
        if (k == n) {
            for (long e : a) {
                ans = ma.mul(ans, e);
            }
            out.append(ma.mod(ans));
            return;
        }
        int mc = 0;
        int pc = 0;
        int zc = 0;
        for (int i = 0; i < n; i++) {
            if (a[i] > 0) {
                pc++;
            } else if (a[i] < 0) {
                mc++;
            } else {
                zc++;
            }
        }
        if (pc == 0) {
            RadixSort.sort(a);
            if (k % 2 == 0) {
                for (int i = 0; i < k; i++) {
                    ans = ma.mul(ans, a[i]);
                }
                out.append(ma.mod(ans));
                return;
            } else {
                for (int i = 0; i < k; i++) {
                    ans = ma.mul(ans, a[n - 1 - i]);
                }
                out.append(ma.mod(ans));
                return;
            }
        }
        if (n - zc < k) {
            out.append(0);
            return;
        }
        ComparativeMergeSort.sort(a, (u, v) -> Long.compare(abs(v), abs(u)));
        int neg = 0;
        long mneg = -1;
        for (int i = 0; i < k; i++) {
            ans = ma.mul(ans, a[i]);
            if (a[i] < 0) {
                neg++;
                mneg = a[i];
            }
        }
        if (neg % 2 == 0) {
            out.append(ma.mod(ans));
            return;
        }
        ans = -ma.div(ans, -mneg);
        for (int i = k; i < n; i++) {
            if (a[i] >= 0) {
                ans = ma.mul(ans, a[i]);
                out.append(ma.mod(ans));
                return;
            }
        }
    }
}



class LongSemiRing extends LongExtendedMagma {
    public LongSemiRing(final LongMonoid addition, final LongMonoid multiplication) {
        super(addition, multiplication, TypicalPropertySets.SEMI_RING);
        addition.requireProperties(EnumSet.of(Property.COMMUTATIVE));
    }
    public static LongSemiRing of(LongExtendedMagma m) {
        if (!m.additiveProps.containsAll(TypicalPropertySets.COMMUTATIVE_MONOID.getProperties())) {
            throw new ClassCastException("addition does not meet the conditions of commutative monoid.");
        }
        LongCommutativeMonoid add = new LongCommutativeMonoid(m::add, m.e0);
        LongMonoid mul = new LongMonoid(m::mul, m.e1);
        return new LongSemiRing(add, mul);
    }
}

final class InsertionSort {
    static void sort(final int[] a) {sort(a, 0, a.length);}
    static void sort(final int[] a, final int from, final int to) {
        for (int i = from + 1; i < to; i++) {
            final int tmp = a[i];
            if (a[i - 1] > tmp) {
                int j = i;
                do {a[j] = a[j - 1]; j--;} while (j > from && a[j - 1] > tmp);
                a[j] = tmp;
            }
        }
    }
    static void sort(final long[] a) {sort(a, 0, a.length);}
    static void sort(final long[] a, final int from, final int to) {
        for (int i = from + 1; i < to; i++) {
            final long tmp = a[i];
            if (a[i - 1] > tmp) {
                int j = i;
                do {a[j] = a[j - 1]; j--;} while (j > from && a[j - 1] > tmp);
                a[j] = tmp;
            }
        }
    }
}



class LongMonoid extends LongMagma implements LongUnital {
    public final long e;
    public LongMonoid(final LongBinaryOperator operator, final long e) {
        super(operator, TypicalPropertySets.MONOID);
        this.e = e;
    }
    @Override
    public long identityElement() {return e;}
    public static <M extends LongMagma & LongUnital> LongMonoid of(M m) {
        m.requireProperties(TypicalPropertySets.MONOID);
        return new LongMonoid(m.operator, m.identityElement());
    }
}



/**
 * @author https://atcoder.jp/users/suisen
 */
final class ModuloArithmetic extends LongField {
    public final long MOD;
    public final LongAbelianGroup addition;
    public final LongAbelianGroup multiplication;
    /**
     * support modulo p arithmetic
     * @param p p s.t. p is a prime number.
     */
    public ModuloArithmetic(final long p) {
        super(
            new LongAbelianGroup((u, v) -> add(u, v, p), 0l, x -> -x),
            new LongAbelianGroup((u, v) -> mul(u, v, p), 1l, x -> inv(x, p)));
        this.addition = new LongAbelianGroup((u, v) -> add(u, v, p), 0l, x -> -x);
        this.multiplication = new LongAbelianGroup((u, v) -> mul(u, v, p), 1l, x -> inv(x, p));
        this.MOD = p;
    }
    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a + b mod p.
     */
    private static long add(final long a, final long b, final long mod) {
        final long s = a + b; return s < 0 ? s + mod : s >= mod ? s - mod : s;
    }
    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a * b mod p.
     */
    private static long mul(final long a, final long b, final long mod) {
        final long ret = (a * b) % mod; return ret < 0 ? ret + mod : ret;
    }
    /**
     * Calculate the value b s.t. a*b mod MOD = 1.
     */
    private static long inv(long a, final long mod) {
        long b = mod;
        long u = 1, v = 0;
        while (b >= 1) {
            final long t = a / b;
            a -= t * b;
            final long tmp1 = a; a = b; b = tmp1;
            u -= t * v;
            final long tmp2 = u; u = v; v = tmp2;
        }
        u %= mod;
        return u < 0 ? u + mod : u;
    }
    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a mod p.
     */
    public long mod(long a) {a %= MOD; return a < 0 ? a + MOD : a;}
    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a_1 + 1_2 + ... + a_k mod p.
     */
    public long sum(final long... a) {
        long ret = 0; for (final long c : a) ret += c;
        return mod(ret);
    }
    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a - b mod p.
     */
    public long sub(final long a, final long b) {
        final long s = a - b; return s < 0 ? s + MOD : s >= MOD ? s - MOD : s;
    }
    /**
     * Calculate x s.t. 0 <= x < MOD /\ x = a_1 * 1_2 * ... * a_k mod p.
     */
    public long prod(final long... a) {
        long ret = 1; for (final long c : a) ret = (ret * c) % MOD;
        return ret < 0 ? ret + MOD : ret;
    }
    /**
     * Calculate x s.t. 0 <= x < MOD /\ b * x = a mod p.
     */
    public long div(final long a, final long b) {return mul(a, inv(b));}
    /**
     * Calculate the value b s.t. a*b mod MOD = 1.
     */
    public long inv(long a) {
        long b = MOD;
        long u = 1, v = 0;
        while (b >= 1) {
            final long t = a / b;
            a -= t * b;
            final long tmp1 = a; a = b; b = tmp1;
            u -= t * v;
            final long tmp2 = u; u = v; v = tmp2;
        }
        return mod(u);
    }
    /**
     * Calculate the permutation nPr.
     */
    public long perm(final long n, final long r) {
        if (n < r) return 0;
        long ret = 1; for (long i = n; i > n - r; i--) ret = mul(ret, i);
        return ret;
    }
    public long perm(final int n, final int r, final long[] fact, final long[] ifact) {
        if (n < r) return 0;
        return mul(fact[n], ifact[n - r]);
    }
    /**
     * Calculate the combination nCr.
     */
    public long comb(long n, long r) {
        if (n < r) return 0;
        r = Math.min(r, n - r);
        n %= MOD;
        long res = 1; for (long d = 1; d <= r; d++) res = div(mul(res, n--), d);
        return res;
    }
    public long comb(final int n, final int r, final long[] fact, final long[] ifact) {
        if (n < r) return 0;
        return prod(fact[n], ifact[n - r], ifact[r]);
    }
    /**
     * Calculate a^b (mod {@code MOD}) in O(log b) time.
     */
    public long pow(final long a, final long b) {
        if (b == 0 || a == 1) return 1;
        final long half = pow(a, b >> 1);
        return (b & 1) == 0 ? mul(half, half) : prod(half, half, a);
    }
    /**
     * calculate x := log_a b mod MOD. (i.e. a^x = b mod MOD). if not exists, return
     * negative value.
     */
    public long log(final long a, final long b) {return log(a, b, getMapForLogarithm(a));}
    public long log(long a, long b, final HashMap<Long, Long> aMap) {
        a %= MOD; b %= MOD;
        final long sq = (long) (Math.sqrt(MOD) + 2);
        final long inv = pow(inv(a), sq);
        for (long p = 0, r = b; p <= sq; p++) {
            if (aMap.containsKey(r)) {
                final long res = p * sq + aMap.get(r);
                if (res > 0) return res;
            }
            r = mul(r, inv);
        }
        return -1;
    }
    public HashMap<Long, Long> getMapForLogarithm(long a) {
        a %= MOD;
        final long sq = (long) (Math.sqrt(MOD) + 2);
        final HashMap<Long, Long> map = new HashMap<>();
        for (long i = 0, ar = 1; i < sq; i++) {
            map.putIfAbsent(ar, i);
            ar = mul(ar, a);
        }
        return map;
    }
    /**
     * build factorial array from 0 to n (closed)
     */
    public long[] factorialArray(final int n) {
        final long[] fact = new long[n + 1];
        fact[0] = fact[1] = 1;
        for (int i = 1; i <= n; i++) fact[i] = mul(fact[i - 1], i);
        return fact;
    }
    public long[] inversiveFactrialArray(final long[] fact) {
        int n = fact.length - 1;
        final long[] ifact = new long[n + 1];
        ifact[n] = inv(fact[n]);
        for (int i = n - 1; i >= 0; i--) ifact[i] = mul(ifact[i + 1], i + 1);
        return ifact;
    }
    /**
     * build power array based on a from 0 to n (closed)
     * @param a base of power
     * @param n
     */
    public long[] powerArray(final long a, final int n) {
        if (a == 2) return powerArray(n);
        final long[] pow = new long[n + 1];
        pow[0] = 1; for (int i = 0; i <= n; i++) pow[i] = mul(pow[i - 1], a);
        return pow;
    }
    /**
     * build power array based on 2 from 0 to n (closed)
     * @param n
     */
    public long[] powerArray(final int n) {
        final long[] pow = new long[n + 1];
        pow[0] = 1;
        for (int i = 1; i <= n; i++) {
            pow[i] = pow[i - 1] << 1;
            if (pow[i] >= MOD) pow[i] -= MOD;
        }
        return pow;
    }
}



abstract class LongMagma extends PropertyHolder {
    protected final LongBinaryOperator operator;
    public LongMagma(final LongBinaryOperator operator, final EnumSet<Property> properties) {
        super(properties);
        this.operator = operator;
    }
    public LongMagma(final LongBinaryOperator operator, final TypicalPropertySets properties) {
        super(properties);
        this.operator = operator;
    }
    public final long apply(final long left, final long right) {return operator.applyAsLong(left, right);}
}



abstract class LongExtendedMagma extends PropertyHolder {
    public final long e0, e1;
    private final LongBinaryOperator addition, multiplication;
    protected final EnumSet<Property> additiveProps, multiplicativeProps;
    public LongExtendedMagma(LongMonoid addition, LongMonoid multiplication, EnumSet<Property> properties) {
        super(properties);
        this.addition = addition.operator;
        this.multiplication = multiplication.operator;
        this.e0 = addition.identityElement();
        this.e1 = multiplication.identityElement();
        this.additiveProps = addition.getProperties();
        this.multiplicativeProps = multiplication.getProperties();
    }
    public LongExtendedMagma(LongMonoid addition, LongMonoid multiplication, TypicalPropertySets properties) {
        this(addition, multiplication, properties.getProperties());
    }
    public final long add(long t, long u) {return addition.applyAsLong(t, u);}
    public final long mul(long t, long u) {return multiplication.applyAsLong(t, u);}
}

/**
 * @author https://atcoder.jp/users/suisen
 */
final class RadixSort {
    private static final int INT_INSERTION_SORT_THRESHOLD = 120;
    private static final int LONG_INSERTION_SORT_THRESHOLD = 250;
    public static void sort(final int[] a) {sort(a, 0, a.length);}
    public static void sortDescending(final int[] a) {sortDescending(a, 0, a.length);}
    public static void sort(final int[] a, final int fr, final int to) {
        if (to - fr <= INT_INSERTION_SORT_THRESHOLD) {InsertionSort.sort(a, fr, to); return;}
        final int n = a.length;
        final int[] c0 = new int[0x101], c1 = new int[0x101], c2 = new int[0x101], c3 = new int[0x101];
        c0[0] = fr; c1[0] = fr; c2[0] = fr; c3[0] = fr;
        for (int i = fr; i < to; i++) {
            final int v = a[i];
            final int b0 = v & 0xff, b1 = v >> 8 & 0xff, b2 = v >> 16 & 0xff, b3 = v >>> 24 ^ 0x80;
            c0[b0 + 1]++; c1[b1 + 1]++; c2[b2 + 1]++; c3[b3 + 1]++;
        }
        for (int i = 0; i < 0x100; i++) {
            c0[i + 1] += c0[i]; c1[i + 1] += c1[i]; c2[i + 1] += c2[i]; c3[i + 1] += c3[i];
        }
        final int[] b = new int[n];
        for (int i = fr; i < to; i++) {final int v = a[i]; b[c0[v        & 0xff]++] = v;}
        for (int i = fr; i < to; i++) {final int v = b[i]; a[c1[v >>> 8  & 0xff]++] = v;}
        for (int i = fr; i < to; i++) {final int v = a[i]; b[c2[v >>> 16 & 0xff]++] = v;}
        for (int i = fr; i < to; i++) {final int v = b[i]; a[c3[v >>> 24 ^ 0x80]++] = v;}
    }
    public static void sortDescending(final int[] a, final int fr, final int to) {
        sort(a, fr, to);
        int l = fr, r = to - 1;
        while (l < r) {final int tmp = a[l]; a[l] = a[r]; a[r] = tmp; l++; r--;}
    }
    public static void sort(final long[] a) {sort(a, 0, a.length);}
    public static void sortDescending(final long[] a) {sortDescending(a, 0, a.length);}
    public static void sort(final long[] a, final int fr, final int to) {
        if (to - fr <= LONG_INSERTION_SORT_THRESHOLD) {InsertionSort.sort(a, fr, to); return;}
        final int n = a.length;
        final int[] c0 = new int[0x101], c1 = new int[0x101], c2 = new int[0x101], c3 = new int[0x101];
        final int[] c4 = new int[0x101], c5 = new int[0x101], c6 = new int[0x101], c7 = new int[0x101];
        c0[0] = fr; c1[0] = fr; c2[0] = fr; c3[0] = fr; c4[0] = fr; c5[0] = fr; c6[0] = fr; c7[0] = fr;
        for (int i = fr; i < to; i++) {
            final long v = a[i];
            final int b0 = (int) (v & 0xff), b1 = (int) (v >> 8 & 0xff), b2 = (int) (v >> 16 & 0xff), b3 = (int) (v >> 24 & 0xff);
            final int b4 = (int) (v >> 32 & 0xff), b5 = (int) (v >> 40 & 0xff), b6 = (int) (v >> 48 & 0xff), b7 = (int) (v >>> 56 ^ 0x80);
            c0[b0 + 1]++; c1[b1 + 1]++; c2[b2 + 1]++; c3[b3 + 1]++; c4[b4 + 1]++; c5[b5 + 1]++; c6[b6 + 1]++; c7[b7 + 1]++;
        }
        for (int i = 0; i < 0x100; i++) {
            c0[i + 1] += c0[i]; c1[i + 1] += c1[i]; c2[i + 1] += c2[i]; c3[i + 1] += c3[i];
            c4[i + 1] += c4[i]; c5[i + 1] += c5[i]; c6[i + 1] += c6[i]; c7[i + 1] += c7[i];
        }
        final long[] b = new long[n];
        for (int i = fr; i < to; i++) {final long v = a[i]; b[c0[(int) (v        & 0xff)]++] = v;}
        for (int i = fr; i < to; i++) {final long v = b[i]; a[c1[(int) (v >>> 8  & 0xff)]++] = v;}
        for (int i = fr; i < to; i++) {final long v = a[i]; b[c2[(int) (v >>> 16 & 0xff)]++] = v;}
        for (int i = fr; i < to; i++) {final long v = b[i]; a[c3[(int) (v >>> 24 & 0xff)]++] = v;}
        for (int i = fr; i < to; i++) {final long v = a[i]; b[c4[(int) (v >>> 32 & 0xff)]++] = v;}
        for (int i = fr; i < to; i++) {final long v = b[i]; a[c5[(int) (v >>> 40 & 0xff)]++] = v;}
        for (int i = fr; i < to; i++) {final long v = a[i]; b[c6[(int) (v >>> 48 & 0xff)]++] = v;}
        for (int i = fr; i < to; i++) {final long v = b[i]; a[c7[(int) (v >>> 56 ^ 0x80)]++] = v;}
    }
    public static void sortDescending(final long[] a, final int fr, final int to) {
        sort(a, fr, to);
        int l = fr, r = to - 1;
        while (l < r) {final long tmp = a[l]; a[l] = a[r]; a[r] = tmp; l++; r--;}
    }
}

@FunctionalInterface
interface LongComparator {
    public int compare(long i, long j);
    public default boolean eq(final long i, final long j) {return compare(i, j) == 0;}
    public default boolean ne(final long i, final long j) {return compare(i, j) != 0;}
    public default boolean gt(final long i, final long j) {return compare(i, j) > 0;}
    public default boolean ge(final long i, final long j) {return compare(i, j) >= 0;}
    public default boolean lt(final long i, final long j) {return compare(i, j) < 0;}
    public default boolean le(final long i, final long j) {return compare(i, j) <= 0;}
}



class LongField extends LongRing {
    public LongField(final LongAbelianGroup addition, final LongMonoid multiplication) {
        super(addition, multiplication);
        multiplication.requireProperties(EnumSet.of(Property.COMMUTATIVE));
    }
}



class LongCommutativeMonoid extends LongMonoid {
    public LongCommutativeMonoid(final LongBinaryOperator operator, final long e) {
        super(operator, e);
        addProperties(Property.COMMUTATIVE);
    }
    public static <M extends LongMagma & LongUnital> LongCommutativeMonoid of(M m) {
        m.requireProperties(TypicalPropertySets.COMMUTATIVE_MONOID.getProperties());
        return new LongCommutativeMonoid(m.operator, m.identityElement());
    }
}


final class ComparativeMergeSort {
    private static int INSERTION_SORT_THRESHOLD = 60;
    public static void sort(final int[] a, final IntComparator comparator) {sort(a, 0, a.length, comparator);}
    public static void sort(final int[] a, final int begin, final int end, final IntComparator comparator) {
        for (int i = begin;;) {
            final int j = i + INSERTION_SORT_THRESHOLD;
            if (j < end) ComparativeInsertionSort.sort(a, i, j, comparator);
            else {ComparativeInsertionSort.sort(a, i, end, comparator); break;}
            i = j;
        }
        final int len = end - begin;
        final int[] work = new int[len];
        for (int block = INSERTION_SORT_THRESHOLD; block <= len; block <<= 1) {
            final int twoBlocks = block << 1;
            for (int from = begin, max = end - block; from < max; from += twoBlocks) {
                final int mid = from + block, to = Math.min(from + twoBlocks, end);
                System.arraycopy(a, from, work, 0, block);
                for (int i = from, wi = 0, ti = mid;; i++) {
                    if (ti == to) {System.arraycopy(work, wi, a, i, block - wi); break;}
                    if (comparator.gt(work[wi], a[ti])) a[i] = a[ti++];
                    else {a[i] = work[wi++]; if (wi == block) break;}
                }
            }
        }
    }
    public static void sort(final long[] a, final LongComparator comparator) {sort(a, 0, a.length, comparator);}
    public static void sort(final long[] a, final int begin, final int end, final LongComparator comparator) {
        for (int i = begin;;) {
            final int j = i + INSERTION_SORT_THRESHOLD;
            if (j < end) ComparativeInsertionSort.sort(a, i, j, comparator);
            else {ComparativeInsertionSort.sort(a, i, end, comparator); break;}
            i = j;
        }
        final int len = end - begin;
        final long[] work = new long[len];
        for (int block = INSERTION_SORT_THRESHOLD; block <= len; block <<= 1) {
            final int twoBlocks = block << 1;
            for (int from = begin, max = end - block; from < max; from += twoBlocks) {
                final int mid = from + block, to = Math.min(from + twoBlocks, end);
                System.arraycopy(a, from, work, 0, block);
                for (int i = from, wi = 0, ti = mid;; i++) {
                    if (ti == to) {System.arraycopy(work, wi, a, i, block - wi); break;}
                    if (comparator.gt(work[wi], a[ti])) a[i] = a[ti++];
                    else {a[i] = work[wi++]; if (wi == block) break;}
                }
            }
        }
    }
}


class LongRing extends LongSemiRing {
    private final LongUnaryOperator additiveInverse;
    public LongRing(final LongAbelianGroup addition, final LongMonoid multiplication) {
        super(addition, multiplication);
        this.additiveInverse = addition::inverse;
    }
    public final long additiveInverse(final long t) {return additiveInverse.applyAsLong(t);}
}


/**
 * @author https://atcoder.jp/users/suisen
 */
final class In {
    public static final FastScanner fsc = new FastScanner();
    public static int ni() {return fsc.nextInt();}
    public static int[] ni(final int n) {
        final int[] a = new int[n];
        for (int i = 0; i < n; i++) a[i] = fsc.nextInt();
        return a;
    }
    public static int[] ni(final int n, final IntUnaryOperator f) {
        final int[] a = new int[n];
        for (int i = 0; i < n; i++) a[i] = f.applyAsInt(fsc.nextInt());
        return a;
    }
    public static int[][] ni(final int n, final int m) {
        final int[][] a = new int[n][m];
        for (int i = 0; i < n; i++) a[i] = ni(m);
        return a;
    }
    public static int[][] ni(final int n, final int m, final IntUnaryOperator f) {
        final int[][] a = new int[n][m];
        for (int i = 0; i < n; i++) a[i] = ni(m, f);
        return a;
    }
    public static long nl() {return fsc.nextLong();}
    public static long[] nl(final int n) {
        final long[] a = new long[n];
        for (int i = 0; i < n; i++) a[i] = fsc.nextLong();
        return a;
    }
    public static long[] nl(final int n, final LongUnaryOperator f) {
        final long[] a = new long[n];
        for (int i = 0; i < n; i++) a[i] = f.applyAsLong(fsc.nextLong());
        return a;
    }
    public static long[][] nl(final int n, final int m) {
        final long[][] a = new long[n][m];
        for (int i = 0; i < n; i++) a[i] = nl(m);
        return a;
    }
    public static long[][] nl(final int n, final int m, final LongUnaryOperator f) {
        final long[][] a = new long[n][m];
        for (int i = 0; i < n; i++) a[i] = nl(m, f);
        return a;
    }
    public static char[] nc() {return fsc.next().toCharArray();}
    public static char[][] nc(final int n) {
        final char[][] c = new char[n][];
        for (int i = 0; i < n; i++) c[i] = nc();
        return c;
    }
    public static double nd() {return fsc.nextDouble();}
    public static double[] nd(final int n) {
        final double[] a = new double[n];
        for (int i = 0; i < n; i++) a[i] = fsc.nextDouble();
        return a;
    }
    public static double[][] nd(final int n, final int m) {
        final double[][] a = new double[n][m];
        for (int i = 0; i < n; i++) a[i] = nd(m);
        return a;
    }
    public static String ns() {return fsc.next();}
    public static String[] ns(final int n) {
        final String[] s = new String[n];
        for (int i = 0; i < n; i++) s[i] = fsc.next();
        return s;
    }
    public static boolean[][] grid(final int h, final int w, final char trueCharacter) {
        final boolean[][] grid = new boolean[h][w];
        for (int i = 0; i < h; i++) {
            final char[] s = fsc.next().toCharArray();
            for (int j = 0; j < w; j++) grid[i][j] = s[j] == trueCharacter;
        }
        return grid;
    }
}


final class FastScanner {
    private final InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;
    private boolean hasNextByte() {
        if (ptr < buflen) return true;
        ptr = 0;
        try {buflen = in.read(buffer);}
        catch (final IOException e) {e.printStackTrace();}
        return buflen > 0;
    }
    private int readByte() {return hasNextByte() ? buffer[ptr++] : -1;}
    public boolean hasNext() {
        while (hasNextByte() && !(33 <= buffer[ptr] && buffer[ptr] <= 126)) ptr++;
        return hasNextByte();
    }
    public String next() {
        if (!hasNext()) throw new NoSuchElementException();
        final StringBuilder sb = new StringBuilder();
        int b = readByte();
        while (33 <= b && b <= 126) {sb.appendCodePoint(b); b = readByte();}
        return sb.toString();
    }
    public long nextLong() {
        if (!hasNext()) throw new NoSuchElementException();
        long n = 0;
        boolean minus = false;
        int b = readByte();
        if (b == '-') {minus = true; b = readByte();}
        if (b < '0' || '9' < b) throw new NumberFormatException();
        for (; ; b = readByte()) {
            if ('0' <= b && b <= '9') n = n * 10 + b - '0';
            else if (b == -1 || !(33 <= b && b <= 126)) return minus ? -n : n;
            else throw new NumberFormatException();
        }
    }
    public int nextInt() {return Math.toIntExact(nextLong());}
    public double nextDouble() {return Double.parseDouble(next());}
}

@FunctionalInterface
interface IntComparator {
    public int compare(int i, int j);
    public default boolean eq(final int i, final int j) {return compare(i, j) == 0;}
    public default boolean ne(final int i, final int j) {return compare(i, j) != 0;}
    public default boolean gt(final int i, final int j) {return compare(i, j) > 0;}
    public default boolean ge(final int i, final int j) {return compare(i, j) >= 0;}
    public default boolean lt(final int i, final int j) {return compare(i, j) < 0;}
    public default boolean le(final int i, final int j) {return compare(i, j) <= 0;}
}

enum Property {ANNIHILATED_BY_ZERO, ASSOCIATIVE, CANCELLATIVE, COMMUTATIVE, DISTRIBUTIVE, IDEMPOTENT;}


enum TypicalPropertySets {
    ABELIAN_GROUP(EnumSet.of(Property.ASSOCIATIVE, Property.CANCELLATIVE, Property.COMMUTATIVE)),
    BOUNDED_SEMI_LATTICE(EnumSet.of(Property.ASSOCIATIVE, Property.COMMUTATIVE, Property.IDEMPOTENT)),
    COMMUTATIVE_MONOID(EnumSet.of(Property.ASSOCIATIVE, Property.COMMUTATIVE)),
    GROUP(EnumSet.of(Property.ASSOCIATIVE, Property.CANCELLATIVE)),
    MONOID(EnumSet.of(Property.ASSOCIATIVE)),
    SEMI_GROUP(EnumSet.of(Property.ASSOCIATIVE)),
    SEMI_LATTICE(EnumSet.of(Property.ASSOCIATIVE, Property.IDEMPOTENT, Property.COMMUTATIVE)),
    SEMI_RING(EnumSet.of(Property.DISTRIBUTIVE, Property.ANNIHILATED_BY_ZERO)),
    RING(EnumSet.of(Property.DISTRIBUTIVE, Property.ANNIHILATED_BY_ZERO));
    private final EnumSet<Property> properties;
    private TypicalPropertySets(EnumSet<Property> properties) {this.properties = properties;}
    public final EnumSet<Property> getProperties() {return properties.clone();}
}

interface LongInversible {long inverse(long t);}

/**
 * @author https://atcoder.jp/users/suisen
 */
class Const {
    public static final long MOD7 = 1_000_000_007;
    public static final long MOD9 = 1_000_000_009;
    public static final long MOD99 = 998_244_353;

    public static final long LINF = Long.MAX_VALUE >> 2;
    public static final int IINF = Integer.MAX_VALUE >> 1;
    public static final double DINF = 1e150;

    public static final double SDELTA = 1e-12;
    public static final double DELTA = 1e-9;
    public static final double LDELTA = 1e-6;

    public static final int[] dx8 = {1, 0, -1, 0, 1, -1, -1, 1};
    public static final int[] dy8 = {0, 1, 0, -1, 1, 1, -1, -1};
    public static final int[] dx4 = {1, 0, -1, 0};
    public static final int[] dy4 = {0, 1, 0, -1};

    private Const(){}
}



class LongAbelianGroup extends LongGroup {
    public LongAbelianGroup(final LongBinaryOperator operator, final long e, final LongUnaryOperator inverse) {
        super(operator, e, inverse);
        addProperties(Property.COMMUTATIVE);
    }
}

interface LongUnital {long identityElement();}



class LongGroup extends LongMonoid implements LongInversible {
    private final LongUnaryOperator inverse;
    public LongGroup(final LongBinaryOperator operator, final long e, final LongUnaryOperator inverse) {
        super(operator, e);
        addProperties(Property.CANCELLATIVE);
        this.inverse = inverse;
    }
    @Override
    public long inverse(final long t) {return inverse.applyAsLong(t);}
}


final class ComparativeInsertionSort {
    static void sort(final int[] a, final int from, final int to, final IntComparator comparator) {
        for (int i = from + 1; i < to; i++) {
            final int tmp = a[i];
            if (comparator.gt(a[i - 1], tmp)) {
                int j = i;
                do {a[j] = a[j - 1]; j--;} while (j > from && comparator.gt(a[j - 1], tmp));
                a[j] = tmp;
            }
        }
    }
    static void sort(final long[] a, final int from, final int to, final LongComparator comparator) {
        for (int i = from + 1; i < to; i++) {
            final long tmp = a[i];
            if (comparator.gt(a[i - 1], tmp)) {
                int j = i;
                do {a[j] = a[j - 1]; j--;} while (j > from && comparator.gt(a[j - 1], tmp));
                a[j] = tmp;
            }
        }
    }
}


abstract class PropertyHolder {
    private final EnumSet<Property> properties;
    public PropertyHolder(final EnumSet<Property> properties) {this.properties = properties;}
    public PropertyHolder(final TypicalPropertySets typicalSet) {this(typicalSet.getProperties());}
    public final EnumSet<Property> getProperties() {return properties;}
    public final void addProperties(final Property... props) {for (Property prop : props) properties.add(prop);}
    public final void addProperties(final EnumSet<Property> props) {for (Property prop : props) properties.add(prop);}
    public final void addProperties(final TypicalPropertySets props) {addProperties(props.getProperties());}
    public final boolean hasProperties(final Property... props) {
        for (final Property prop : props) if (!properties.contains(prop)) return false;
        return true;
    }
    public final boolean hasProperties(final EnumSet<Property> props) {
        for (final Property prop : props) if (!properties.contains(prop)) return false;
        return true;
    }
    public final boolean hasProperties(final TypicalPropertySets props) {return hasProperties(props.getProperties());}
    public final void requireProperties(final EnumSet<Property> props, final Property... ps) {
        final ArrayList<Property> unmet = new ArrayList<>();
        for (final Property prop : props) if (!properties.contains(prop)) unmet.add(prop);
        for (final Property prop : ps) if (!properties.contains(prop)) unmet.add(prop);
        if (unmet.size() > 0) throw new UnsatisfiedRequiredPropertiesException(unmet);
    }
    public final void requireProperties(final TypicalPropertySets props, final Property... ps) {
        requireProperties(props.getProperties(), ps);
    }
    private static final class UnsatisfiedRequiredPropertiesException extends RuntimeException {
        private static final long serialVersionUID = 1L;
        private UnsatisfiedRequiredPropertiesException(final Collection<Property> props) {
            for (final Property prop : props) System.err.println(prop.name() + "should be satisfied.");
        }
    }
}