Problem:
Score : 500 points

Problem Statement

#nck {
width: 30px;
height: auto;
}

A cheetah and a cheater are going to play the game of Nim.
In this game they use N piles of stones.
Initially the i-th pile contains a_i stones.
The players take turns alternately, and the cheetah plays first.
In each turn, the player chooses one of the piles, and takes one or more stones from the pile.
The player who can't make a move loses.

However, before the game starts, the cheater wants to cheat a bit to make sure that he can win regardless of the moves by the cheetah.
From each pile, the cheater takes zero or one stone and eats it before the game.
In case there are multiple ways to guarantee his winning, he wants to minimize the number of stones he eats.

Compute the number of stones the cheater will eat.
In case there is no way for the cheater to win the game even with the cheating, print -1 instead.

Constraints

1 ≤ N ≤ 10^5

2 ≤ a_i ≤ 10^9

Input

The input is given from Standard Input in the following format:

N
a_1
:
a_N

Output

Print the answer.

Sample Input 1

3
2
3
4

Sample Output 1

3

The only way for the cheater to win the game is to take stones from all piles and eat them.

Sample Input 2

3
100
100
100

Sample Output 2

-1

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)


import java.io.*;

import static java.lang.Math.*;
import static java.lang.Math.min;

import java.math.BigDecimal;
import java.util.*;

import java.util.stream.*;

import static java.util.Arrays.*;


/**
 * @author baito
 */
@SuppressWarnings("unchecked")
public class Main {
    static boolean DEBUG = true;
    static StringBuilder sb = new StringBuilder();
    static int INF = 1234567890;
    static int MINF = -1234567890;
    static long LINF = 123456789123456789L;
    static long MLINF = -123456789123456789L;
    static long MOD = 1000000007;
    static double EPS = 1e-10;
    static int[] y4 = {0, 1, 0, -1};
    static int[] x4 = {1, 0, -1, 0};
    static int[] y8 = {0, 1, 0, -1, -1, 1, 1, -1};
    static int[] x8 = {1, 0, -1, 0, 1, -1, 1, -1};
    static ArrayList<Long> Fa;
    static boolean[] isPrime;
    static ArrayList<Integer> primes;
    static char[][] ban;
    static long maxRes = Long.MIN_VALUE;
    static long minRes = Long.MAX_VALUE;

    static int N;
    static int[] A;

    public static void solve() throws Exception {
        //longを忘れるなオーバーフローするぞ
        N = ni();
        A = nia(N);
        boolean[] exist = new boolean[35];
        for (int i = 0; i < N; i++) {
            for (int j = 0; j < 33; j++) {
                if (bget(A[i], j)) {
                    exist[j] = true;
                    break;
                }
            }
        }
        long xor = 0;
        for (int i = 0; i < N; i++) {
            xor ^= A[i];
        }
        int cou = 0;
        boolean rev = false;
        for (int i = 32; i >= 0; i--) {
            if (bget(xor, i) ^ rev) {
                if (exist[i]) {
                    cou++;
                    rev ^= true;
                } else {
                    System.out.println("-1");
                    return;
                }
            }
        }
        System.out.println(cou);
    }

    public static boolean calc(long va) {
        //貪欲にギリギリセーフを選んでいく。
        int v = (int) va;
        return true;
    }

    //条件を満たす最大値、あるいは最小値を求める
    static int mgr(long ok, long ng) {
        //int ok = 0; //解が存在する
        //int ng = N; //解が存在しない
        while (Math.abs(ok - ng) > 1) {
            long mid;
            if (ok < 0 && ng > 0 || ok > 0 && ng < 0) mid = (ok + ng) / 2;
            else mid = ok + (ng - ok) / 2;

            if (calc(mid)) {
                ok = mid;
            } else {
                ng = mid;
            }
        }
        if (calc(ok)) return (int) ok;
        else return -1;
    }

    static ArrayList<Integer> divisors(int n) {
        ArrayList<Integer> res = new ArrayList<>();
        for (int i = 1; i <= Math.sqrt(n); i++) {
            if (n % i == 0) {
                res.add(i);
                if (i != n / i) res.add(n / i);
            }
        }
        return res;
    }

    static ArrayList<Long> divisors(long n) {
        ArrayList<Long> res = new ArrayList<>();
        for (long i = 1; i <= Math.sqrt(n); i++) {
            if (n % i == 0) {
                res.add(i);
                if (i != n / i) res.add(n / i);
            }
        }
        return res;
    }

    static ArrayList<Integer> factorization(int n) {
        if (primes == null) setPrimes();
        ArrayList<Integer> fact = new ArrayList<>();
        for (int p : primes) {
            if (n % p == 0) fact.add(p);
            while (n % p == 0) n /= p;
            if (n == 1) break;
        }
        if (n != 1) fact.add(n);
        return fact;
    }

    boolean equal(double a, double b) {
        return a == 0 ? abs(b) < EPS : abs((a - b) / a) < EPS;
    }

    public static void matPrint(long[][] a) {
        for (int hi = 0; hi < a.length; hi++) {
            for (int wi = 0; wi < a[0].length; wi++) {
                System.out.print(a[hi][wi] + " ");
            }
            System.out.println("");
        }
    }

    //rにlを掛ける l * r
    public static long[][] matMul(long[][] l, long[][] r) throws IOException {
        int lh = l.length;
        int lw = l[0].length;
        int rh = r.length;
        int rw = r[0].length;
        //lwとrhが,同じである必要がある
        if (lw != rh) throw new IOException();
        long[][] res = new long[lh][rw];
        for (int i = 0; i < lh; i++) {
            for (int j = 0; j < rw; j++) {
                for (int k = 0; k < lw; k++) {
                    res[i][j] = mSum(res[i][j], mMul(l[i][k], r[k][j]));
                }
            }
        }
        return res;
    }

    public static long[][] matPow(long[][] a, int n) throws IOException {
        int h = a.length;
        int w = a[0].length;
        if (h != w) throw new IOException();
        long[][] res = new long[h][h];
        for (int i = 0; i < h; i++) {
            res[i][i] = 1;
        }
        long[][] pow = a.clone();
        while (n > 0) {
            if (bget(n, 0)) res = matMul(pow, res);
            pow = matMul(pow, pow);
            n >>= 1;
        }
        return res;
    }

    public static void chMax(long v) {
        maxRes = Math.max(maxRes, v);
    }

    public static void chMin(long v) {
        minRes = Math.min(minRes, v);
    }

    //便利系
    //左端、右端の情報をもたせる。
    public static void setLR(int[] a, ArrayList<Integer> l, ArrayList<Integer> r) {
        for (int i = 0; i < a.length; i++) {
            if (a[i] > 0) {
                int j = i;
                while (a[j] > 0) j++;
                l.add(i);
                r.add(j);
                i = j - 1;
            }
        }
    }

    public static int[] imosu(int[] f, int[] t, int n) {
        int[] imosu = new int[n + 1];
        for (int i = 0; i < f.length; i++) {
            imosu[f[i]]++;
            imosu[t[i] + 1]--;
        }
        for (int i = 0; i < n; i++) {
            imosu[i + 1] += imosu[i];
        }
        return imosu;
    }

    static int[] inverse(int[] a) {
        int[] res = new int[a.length];
        for (int i = 0; i < a.length; i++)
            res[a[i]] = i;
        return res;
    }

    //2点間の行き先を配列に持たせる
    static int[][] packE(int n, int[] from, int[] to) {
        int[][] g = new int[n][];
        int[] p = new int[n];
        for (int f : from)
            p[f]++;
        for (int t : to)
            p[t]++;
        for (int i = 0; i < n; i++)
            g[i] = new int[p[i]];
        for (int i = 0; i < from.length; i++) {
            g[from[i]][--p[from[i]]] = to[i];
            g[to[i]][--p[to[i]]] = from[i];
        }
        return g;
    }

    public static String notE(double v) {
        return BigDecimal.valueOf(v).toPlainString();
    }

    public static void print(char[][] a) {
        for (int i = 0; i < a.length; i++) {
            for (int j = 0; j < a[0].length; j++) {
                System.out.print(a[i][j]);
            }
            System.out.println("");
        }
    }

    public static <T> void print(ArrayList<T> a) {
        for (T t : a) {
            System.out.println(t);
        }
    }

    public static void print(int[] a) {
        for (int i = 0; i < a.length; i++)
            System.out.println(a[i]);
    }

    public static void print(long[] a) {
        for (int i = 0; i < a.length; i++)
            System.out.println(a[i]);
    }

    public static void print(int[][] a) {
        for (int i = 0; i < a.length; i++) {
            for (int j = 0; j < a[0].length; j++) {
                System.out.print(a[i][j]);
            }
            System.out.println("");
        }
    }

    //bit関連
    public static boolean bget(BitSet bit, int keta) {
        return bit.nextSetBit(keta) == keta;
    }

    public static boolean bget(long bit, int keta) {
        return ((bit >> keta) & 1) == 1;
    }


    public static int restoreHashA(long key) {
        return (int) (key >> 32);
    }

    public static int restoreHashB(long key) {
        return (int) (key & -1);
    }

    //正の数のみ
    public static long getHashKey(int a, int b) {
        return (long) a << 32 | b;
    }
    //数学関係--------------------------------

    //a/bを返す
    public static long ceil(long a, long b) {
        return (a % b == 0) ? a / b : a / b + 1;
    }

    public static long sqrt(long v) {
        long res = (long) Math.sqrt(v);
        while (res * res > v) res--;
        return res;
    }

    static long ncr2(int a, int b) {
        if (b == 0) return 1;
        else if (a < b) return 0;
        long res = 1;
        for (int i = 0; i < b; i++) {
            res *= a - i;
            res /= i + 1;
        }
        return res;
    }

    static double[][] PER_DP;

    static double ncrPer(int n, int r) {
        if (n < r) return 0;
        if (PER_DP == null) {
            PER_DP = new double[5001][5001];
            PER_DP[0][0] = 1;
            for (int ni = 0; ni < PER_DP.length - 1; ni++) {
                for (int ri = 0; ri < ni + 1; ri++) {
                    PER_DP[ni + 1][ri] += PER_DP[ni][ri] / 2;
                    PER_DP[ni + 1][ri + 1] += PER_DP[ni][ri] / 2;
                }
            }
        }
        return PER_DP[n][r];
    }

    public static int mod(int a, int m) {
        return a >= 0 ? a % m : (int) (a + ceil(-a, m) * m) % m;
    }

    static long mNcr(int n, int r) {
        if (n < 0 || r < 0 || n < r) return 0;
        if (Fa == null || Fa.size() <= n) factorial(n);
        long result = Fa.get(n);
        result = mMul(result, mInv(Fa.get(n - r)));
        result = mMul(result, mInv(Fa.get(r)));
        return result;
    }

    public static long mSum(long... lar) {
        long res = 0;
        for (long l : lar)
            res = (res + l % MOD) % MOD;
        return (res + MOD) % MOD;
    }

    public static long mDiff(long... lar) {
        long res = lar[0] % MOD;
        for (int i = 1; i < lar.length; i++) {
            res = (res - lar[i] % MOD) % MOD;
        }
        return (res + MOD) % MOD;
    }

    public static long mMul(long... lar) {
        long res = 1;
        for (long l : lar)
            res = (res * (l % MOD)) % MOD;
        return (res + MOD) % MOD;
    }

    public static long mDiv(long... lar) {
        long res = lar[0] % MOD;
        for (int i = 1; i < lar.length; i++) {
            res = mMul(res, mInv(lar[i]));
        }
        return (res + MOD) % MOD;
    }

    static long mInv(long n) {
        return mPow(n, MOD - 2);
    }

    static long mPow(long x, long n) {
        long res = 1L;
        while (n > 0) {
            if ((n & 1) == 1) {
                res = res * x % MOD;
            }
            x = x * x % MOD;
            n >>= 1;
        }
        return (res + MOD) % MOD;
    }

    static void factorial(int n) {
        if (Fa == null) {
            Fa = new ArrayList<>();
            Fa.add(1L);
            Fa.add(1L);
        }
        for (int i = Fa.size(); i <= n; i++) {
            Fa.add((Fa.get(i - 1) * i) % MOD);
        }
    }


    //↑nCrをmod計算するために必要
    static long lcm(long n, long r) {
        return n / gcd(n, r) * r;
    }

    static int gcd(int n, int r) {
        return r == 0 ? n : gcd(r, n % r);
    }

    static long gcd(long n, long r) {
        return r == 0 ? n : gcd(r, n % r);
    }

    public static int u0(int a) {
        if (a < 0) return 0;
        return a;
    }

    public static long u0(long a) {
        if (a < 0) return 0;
        return a;
    }

    public static boolean[][] tbt(char[][] s, char c) {
        boolean[][] res = new boolean[s.length][s[0].length];
        for (int hi = 0; hi < s.length; hi++)
            for (int wi = 0; wi < s[0].length; wi++)
                if (s[hi][wi] == c) res[hi][wi] = true;
        return res;
    }

    public static int[] tia(int a) {
        int[] res = new int[keta(a)];
        for (int i = res.length - 1; i >= 0; i--) {
            res[i] = a % 10;
            a /= 10;
        }
        return res;
    }

    public static int[][] tit(char[][] a) {
        int[][] res = new int[a.length][a[0].length];
        for (int hi = 0; hi < a.length; hi++) {
            for (int wi = 0; wi < a[0].length; wi++) {
                res[hi][wi] = a[hi][wi] - '0';
            }
        }
        return res;
    }

    public static Integer[] toIntegerArray(int[] ar) {
        Integer[] res = new Integer[ar.length];
        for (int i = 0; i < ar.length; i++) {
            res[i] = ar[i];
        }
        return res;
    }

    public static long bitGetCombSizeK(int k) {
        return (1 << k) - 1;
    }

    //k個の次の組み合わせをビットで返す 大きさに上限はない 110110 -> 111001
    public static long bitNextComb(long comb) {
        long x = comb & -comb; //最下位の1
        long y = comb + x; //連続した下の1を繰り上がらせる
        return ((comb & ~y) / x >> 1) | y;
    }

    public static int keta(long num) {
        int res = 0;
        while (num > 0) {
            num /= 10;
            res++;
        }
        return res;
    }

    public static int ketaSum(long num) {
        int res = 0;
        while (num > 0) {
            res += num % 10;
            num /= 10;
        }
        return res;
    }

    public static boolean isOutofIndex(int x, int y, int w, int h) {
        if (x < 0 || y < 0) return true;
        if (w <= x || h <= y) return true;
        return false;
    }

    public static boolean isOutofIndex(int x, int y, char[][] ban) {
        if (x < 0 || y < 0) return true;
        if (ban[0].length <= x || ban.length <= y) return true;
        return false;
    }

    public static int arrayCount(int[] a, int v) {
        int res = 0;
        for (int i = 0; i < a.length; i++) {
            if (a[i] == v) res++;
        }
        return res;
    }

    public static int arrayCount(long[] a, int v) {
        int res = 0;
        for (int i = 0; i < a.length; i++) {
            if (a[i] == v) res++;
        }
        return res;
    }

    public static int arrayCount(int[][] a, int v) {
        int res = 0;
        for (int hi = 0; hi < a.length; hi++) {
            for (int wi = 0; wi < a[0].length; wi++) {
                if (a[hi][wi] == v) res++;
            }
        }
        return res;
    }

    public static int arrayCount(long[][] a, int v) {
        int res = 0;
        for (int hi = 0; hi < a.length; hi++) {
            for (int wi = 0; wi < a[0].length; wi++) {
                if (a[hi][wi] == v) res++;
            }
        }
        return res;
    }

    public static int arrayCount(char[][] a, char v) {
        int res = 0;
        for (int hi = 0; hi < a.length; hi++) {
            for (int wi = 0; wi < a[0].length; wi++) {
                if (a[hi][wi] == v) res++;
            }
        }
        return res;
    }

    public static void setPrimes() {
        int n = 100001;
        isPrime = new boolean[n];
        Arrays.fill(isPrime, true);
        isPrime[0] = isPrime[1] = false;
        for (int i = 2; i * i <= n; i++) {
            if (!isPrime[i]) continue;
            for (int j = i * 2; j < n; j += i) {
                isPrime[j] = false;
            }
        }
        primes = new ArrayList<>();
        for (int i = 2; i < n; i++) {
            if (isPrime[i]) primes.add(i);
        }
    }

    public static void revSort(int[] a) {
        Arrays.sort(a);
        reverse(a);
    }

    public static void revSort(long[] a) {
        Arrays.sort(a);
        reverse(a);
    }

    public static P[] clone(P[] ar) {
        P[] res = new P[ar.length];
        for (int i = 0; i < ar.length; i++) {
            res[i] = new P(ar[i].x, ar[i].y);
        }
        return res;
    }

    public static int[][] clone(int[][] ar) {
        int[][] nr = new int[ar.length][ar[0].length];
        for (int i = 0; i < ar.length; i++)
            for (int j = 0; j < ar[0].length; j++)
                nr[i][j] = ar[i][j];
        return nr;
    }

    public static long[][] clone(long[][] ar) {
        long[][] nr = new long[ar.length][ar[0].length];
        for (int i = 0; i < ar.length; i++)
            for (int j = 0; j < ar[0].length; j++)
                nr[i][j] = ar[i][j];
        return nr;
    }

    public static double[][] clone(double[][] ar) {
        double[][] nr = new double[ar.length][ar[0].length];
        for (int i = 0; i < ar.length; i++)
            for (int j = 0; j < ar[0].length; j++)
                nr[i][j] = ar[i][j];
        return nr;
    }

    public static boolean[][] clone(boolean[][] ar) {
        boolean[][] nr = new boolean[ar.length][ar[0].length];
        for (int i = 0; i < ar.length; i++)
            for (int j = 0; j < ar[0].length; j++)
                nr[i][j] = ar[i][j];
        return nr;
    }

    public static char[][] clone(char[][] ar) {
        char[][] nr = new char[ar.length][ar[0].length];
        for (int i = 0; i < ar.length; i++)
            for (int j = 0; j < ar[0].length; j++)
                nr[i][j] = ar[i][j];
        return nr;
    }

    public static int[][][] clone(int[][][] ar) {
        int[][][] nr = new int[ar.length][ar[0].length][ar[0][0].length];
        for (int i = 0; i < ar.length; i++)
            for (int j = 0; j < ar[0].length; j++)
                for (int k = 0; k < ar[0][0].length; k++)
                    nr[i][j][k] = ar[i][j][k];
        return nr;
    }

    public static long[][][] clone(long[][][] ar) {
        long[][][] nr = new long[ar.length][ar[0].length][ar[0][0].length];
        for (int i = 0; i < ar.length; i++)
            for (int j = 0; j < ar[0].length; j++)
                for (int k = 0; k < ar[0][0].length; k++)
                    nr[i][j][k] = ar[i][j][k];
        return nr;
    }

    public static double[][][] clone(double[][][] ar) {
        double[][][] nr = new double[ar.length][ar[0].length][ar[0][0].length];
        for (int i = 0; i < ar.length; i++)
            for (int j = 0; j < ar[0].length; j++)
                for (int k = 0; k < ar[0][0].length; k++)
                    nr[i][j][k] = ar[i][j][k];
        return nr;
    }

    public static boolean[][][] clone(boolean[][][] ar) {
        boolean[][][] nr = new boolean[ar.length][ar[0].length][ar[0][0].length];
        for (int i = 0; i < ar.length; i++)
            for (int j = 0; j < ar[0].length; j++)
                for (int k = 0; k < ar[0][0].length; k++)
                    nr[i][j][k] = ar[i][j][k];
        return nr;
    }

    public static char[][][] clone(char[][][] ar) {
        char[][][] nr = new char[ar.length][ar[0].length][ar[0][0].length];
        for (int i = 0; i < ar.length; i++)
            for (int j = 0; j < ar[0].length; j++)
                for (int k = 0; k < ar[0][0].length; k++)
                    nr[i][j][k] = ar[i][j][k];
        return nr;
    }

    /**
     * <h1>指定した値以上の先頭のインデクスを返す</h1>
     * <p>配列要素が０のときは、０が返る。</p>
     *
     * @return<b>int</b> ： 探索した値以上で、先頭になるインデクス
     * 値が無ければ、挿入できる最小のインデックス
     */
    public static <T extends Number> int lowerBound(final List<T> lis, final T value) {
        int low = 0;
        int high = lis.size();
        int mid;

        while (low < high) {
            mid = ((high - low) >>> 1) + low;    //(low + high) / 2 (オーバーフロー対策)
            if (lis.get(mid).doubleValue() < value.doubleValue()) {
                low = mid + 1;
            } else {
                high = mid;
            }
        }
        return low;
    }

    //v未満で最大のiを返す。ただしvを満たすiがあるなら最小のiを返す
    public static <T extends Number> int rlowerBound(final List<T> lis, final T value) {
        int ind = lowerBound(lis, value);
        if (ind == lis.size() || !lis.get(ind).equals(value)) ind--;
        return ind;
    }

    /**
     * <h1>指定した値より大きい先頭のインデクスを返す</h1>
     * <p>配列要素が０のときは、０が返る。</p>
     *
     * @return<b>int</b> ： 探索した値より上で、先頭になるインデクス
     * 値が無ければ、挿入できる最小のインデックス
     */
    public static <T extends Number> int upperBound(final List<T> lis, final T value) {
        int low = 0;
        int high = lis.size();
        int mid;
        while (low < high) {
            mid = ((high - low) >>> 1) + low;    //(low + high) / 2 (オーバーフロー対策)
            if (lis.get(mid).doubleValue() < value.doubleValue()) {
                low = mid + 1;
            } else {
                high = mid;
            }
        }
        return low;
    }

    public static int lowerBound(final int[] arr, final int value) {
        int low = 0;
        int high = arr.length;
        int mid;

        while (low < high) {
            mid = ((high - low) >>> 1) + low;    //(low + high) / 2 (オーバーフロー対策)
            if (arr[mid] < value) {
                low = mid + 1;
            } else {
                high = mid;
            }
        }
        return low;
    }

    public static int rlowerBound(final int[] arr, final int value) {
        int ind = lowerBound(arr, value);
        if (ind == arr.length || arr[ind] != value) ind--;
        return ind;
    }


    public static int upperBound(final int[] arr, final int value) {
        int low = 0;
        int high = arr.length;
        int mid;
        while (low < high) {
            mid = ((high - low) >>> 1) + low;    //(low + high) / 2 (オーバーフロー対策)
            if (arr[mid] <= value) {
                low = mid + 1;
            } else {
                high = mid;
            }
        }
        return low;
    }

    public static int lowerBound(final long[] arr, final long value) {
        int low = 0;
        int high = arr.length;
        int mid;
        while (low < high) {
            mid = ((high - low) >>> 1) + low;    //(low + high) / 2 (オーバーフロー対策)
            if (arr[mid] < value) {
                low = mid + 1;
            } else {
                high = mid;
            }
        }
        return low;
    }

    public static int rlowerBound(final long[] arr, final long value) {
        int ind = lowerBound(arr, value);
        if (ind == arr.length || arr[ind] != value) ind--;
        return ind;
    }


    public static int upperBound(final long[] arr, final long value) {
        int low = 0;
        int high = arr.length;
        int mid;
        while (low < high) {
            mid = ((high - low) >>> 1) + low;    //(low + high) / 2 (オーバーフロー対策)
            if (arr[mid] <= value) {
                low = mid + 1;
            } else {
                high = mid;
            }
        }
        return low;
    }

    //次の順列に書き換える、最大値ならfalseを返す
    public static boolean nextPermutation(int A[]) {
        int len = A.length;
        int pos = len - 2;
        for (; pos >= 0; pos--) {
            if (A[pos] < A[pos + 1]) break;
        }
        if (pos == -1) return false;

        //posより大きい最小の数を二分探索
        int ok = pos + 1;
        int ng = len;
        while (Math.abs(ng - ok) > 1) {
            int mid = (ok + ng) / 2;
            if (A[mid] > A[pos]) ok = mid;
            else ng = mid;

        }

        swap(A, pos, ok);
        reverse(A, pos + 1, len - 1);


        return true;
    }

    //次の順列に書き換える、最小値ならfalseを返す
    public static boolean prevPermutation(int A[]) {
        int len = A.length;
        int pos = len - 2;
        for (; pos >= 0; pos--) {
            if (A[pos] > A[pos + 1]) break;
        }
        if (pos == -1) return false;

        //posより小さい最大の数を二分探索
        int ok = pos + 1;
        int ng = len;
        while (Math.abs(ng - ok) > 1) {
            int mid = (ok + ng) / 2;
            if (A[mid] < A[pos]) ok = mid;
            else ng = mid;

        }

        swap(A, pos, ok);
        reverse(A, pos + 1, len - 1);


        return true;
    }


    static <T> void swap(T[] x, int i, int j) {
        T t = x[i];
        x[i] = x[j];
        x[j] = t;
    }

    static void swap(char[] x, int i, int j) {
        char t = x[i];
        x[i] = x[j];
        x[j] = t;
    }

    static void swap(int[] x, int i, int j) {
        int t = x[i];
        x[i] = x[j];
        x[j] = t;
    }

    public static String reverse(String a) {
        sb.append(a);
        String res = sb.reverse().toString();
        sb.setLength(0);
        return new String(res);
    }

    public static void reverse(int[] x) {
        int l = 0;
        int r = x.length - 1;
        while (l < r) {
            int temp = x[l];
            x[l] = x[r];
            x[r] = temp;
            l++;
            r--;
        }
    }

    public static void reverse(long[] x) {
        int l = 0;
        int r = x.length - 1;
        while (l < r) {
            long temp = x[l];
            x[l] = x[r];
            x[r] = temp;
            l++;
            r--;
        }
    }

    public static void reverse(char[] x) {
        int l = 0;
        int r = x.length - 1;
        while (l < r) {
            char temp = x[l];
            x[l] = x[r];
            x[r] = temp;
            l++;
            r--;
        }
    }

    public static void reverse(int[] x, int s, int e) {
        int l = s;
        int r = e;
        while (l < r) {
            int temp = x[l];
            x[l] = x[r];
            x[r] = temp;
            l++;
            r--;
        }
    }

    static int length(int a) {
        int cou = 0;
        while (a != 0) {
            a /= 10;
            cou++;
        }
        return cou;
    }

    static int length(long a) {
        int cou = 0;
        while (a != 0) {
            a /= 10;
            cou++;
        }
        return cou;
    }

    static int cou(boolean[] a) {
        int res = 0;
        for (boolean b : a) {
            if (b) res++;
        }
        return res;
    }

    static int cou(String s, char c) {
        int res = 0;
        for (char ci : s.toCharArray()) {
            if (ci == c) res++;
        }
        return res;
    }

    static int cou(char[][] a, char c) {
        int co = 0;
        for (int i = 0; i < a.length; i++)
            for (int j = 0; j < a[0].length; j++)
                if (a[i][j] == c) co++;
        return co;
    }

    static int cou(int[] a, int key) {
        int co = 0;
        for (int i = 0; i < a.length; i++)
            if (a[i] == key) co++;
        return co;
    }

    static int cou(long[] a, long key) {
        int co = 0;
        for (int i = 0; i < a.length; i++)
            if (a[i] == key) co++;
        return co;
    }

    static int cou(int[][] a, int key) {
        int co = 0;
        for (int i = 0; i < a.length; i++)
            for (int j = 0; j < a[0].length; j++)
                if (a[i][j] == key) co++;
        return co;
    }

    static int[] couArray(int[] a) {
        int[] res = new int[max(a) + 1];
        for (int i : a) {
            res[i]++;
        }
        return res;
    }

    static void fill(int[][] a, int v) {
        for (int i = 0; i < a.length; i++)
            for (int j = 0; j < a[0].length; j++)
                a[i][j] = v;
    }

    static void fill(char[][] a, char c) {
        for (int i = 0; i < a.length; i++)
            for (int j = 0; j < a[0].length; j++)
                a[i][j] = c;
    }

    static void fill(long[][] a, long v) {
        for (int i = 0; i < a.length; i++)
            for (int j = 0; j < a[0].length; j++)
                a[i][j] = v;
    }

    static void fill(boolean[][] a, boolean v) {
        for (int i = 0; i < a.length; i++)
            for (int j = 0; j < a[0].length; j++)
                a[i][j] = v;
    }

    static void fill(int[][][] a, int v) {
        for (int i = 0; i < a.length; i++)
            for (int j = 0; j < a[0].length; j++)
                for (int k = 0; k < a[0][0].length; k++)
                    a[i][j][k] = v;
    }

    static void fill(long[][][] a, long v) {
        for (int i = 0; i < a.length; i++)
            for (int j = 0; j < a[0].length; j++)
                for (int k = 0; k < a[0][0].length; k++)
                    a[i][j][k] = v;
    }

    static int max(int... a) {
        int res = Integer.MIN_VALUE;
        for (int i : a) {
            res = Math.max(res, i);
        }
        return res;
    }

    static long max(long... a) {
        long res = Integer.MIN_VALUE;
        for (long i : a) {
            res = Math.max(res, i);
        }
        return res;
    }

    static long min(long... a) {
        long res = Long.MAX_VALUE;
        for (long i : a) {
            res = Math.min(res, i);
        }
        return res;
    }

    static int max(int[][] ar) {
        int res = Integer.MIN_VALUE;
        for (int i[] : ar)
            res = Math.max(res, max(i));
        return res;
    }

    static long max(long[][] ar) {
        long res = Integer.MIN_VALUE;
        for (long i[] : ar)
            res = Math.max(res, max(i));
        return res;
    }

    static int min(int... a) {
        int res = Integer.MAX_VALUE;
        for (int i : a) {
            res = Math.min(res, i);
        }
        return res;
    }


    static int min(int[][] ar) {
        int res = Integer.MAX_VALUE;
        for (int i[] : ar)
            res = Math.min(res, min(i));
        return res;
    }

    public static <T extends Number> long sum(ArrayList<T> lis) {
        long res = 0;
        for (T li : lis) {
            res += li.longValue();
        }
        return res;
    }

    static long sum(int[] a) {
        long cou = 0;
        for (int i : a)
            cou += i;
        return cou;
    }

    static long sum(long[] a) {
        long cou = 0;
        for (long i : a)
            cou += i;
        return cou;
    }


    //FastScanner

    static BufferedReader reader = new BufferedReader(new InputStreamReader(System.in));
    static StringTokenizer tokenizer = null;

    public static String next() {
        if (tokenizer == null || !tokenizer.hasMoreTokens()) {
            try {
                tokenizer = new StringTokenizer(reader.readLine());
            } catch (IOException e) {
                throw new RuntimeException(e);
            }
        }
        return tokenizer.nextToken();
    }

    /*public String nextChar(){
        return (char)next()[0];
    }*/
    public static String nextLine() {
        if (tokenizer == null || !tokenizer.hasMoreTokens()) {
            try {
                return reader.readLine();
            } catch (IOException e) {
                throw new RuntimeException(e);
            }
        }

        return tokenizer.nextToken("\n");
    }

    public static long nl() {
        return Long.parseLong(next());
    }

    public static String n() {
        return next();
    }

    public static int ni() {
        return Integer.parseInt(next());
    }

    public static double nd() {
        return Double.parseDouble(next());
    }

    public static int[] nia(int n) {
        int[] a = new int[n];
        for (int i = 0; i < n; i++) {
            a[i] = ni();
        }
        return a;
    }


    //1-index
    public static int[] niao(int n) {
        int[] a = new int[n + 1];
        for (int i = 1; i < n + 1; i++) {
            a[i] = ni();
        }
        return a;
    }

    //番兵法
    public static int[] nias(int n, int end) {
        int[] a = new int[n + 1];
        for (int i = 0; i < n; i++) {
            a[i] = ni();
        }
        a[n] = end;
        return a;
    }

    public static int[] niad(int n) {
        int[] a = new int[n];
        for (int i = 0; i < n; i++) {
            a[i] = ni() - 1;
        }
        return a;
    }

    public static P[] npa(int n) {
        P[] p = new P[n];
        for (int i = 0; i < n; i++) {
            p[i] = new P(ni(), ni());
        }
        return p;
    }

    public static P[] npad(int n) {
        P[] p = new P[n];
        for (int i = 0; i < n; i++) {
            p[i] = new P(ni() - 1, ni() - 1);
        }
        return p;
    }

    public static int[][] nit(int h, int w) {
        int[][] a = new int[h][w];
        for (int hi = 0; hi < h; hi++) {
            for (int wi = 0; wi < w; wi++) {
                a[hi][wi] = ni();
            }
        }
        return a;
    }

    public static int[][] nitd(int h, int w) {
        int[][] a = new int[h][w];
        for (int hi = 0; hi < h; hi++) {
            for (int wi = 0; wi < w; wi++) {
                a[hi][wi] = ni() - 1;
            }
        }
        return a;
    }

    static int[][] S_ARRAY;
    static long[][] S_LARRAY;
    static int S_INDEX;
    static int S_LINDEX;

    //複数の配列を受け取る
    public static int[] niah(int n, int k) throws Exception {
        if (S_ARRAY == null) {
            S_ARRAY = new int[k][n];
            for (int j = 0; j < n; j++) {
                for (int i = 0; i < k; i++) {
                    S_ARRAY[i][j] = ni();
                }
            }
        }
        return S_ARRAY[S_INDEX++];
    }

    public static long[] nlah(int n, int k) throws Exception {
        if (S_LARRAY == null) {
            S_LARRAY = new long[k][n];
            for (int j = 0; j < n; j++) {
                for (int i = 0; i < k; i++) {
                    S_LARRAY[i][j] = nl();
                }
            }
        }
        return S_LARRAY[S_LINDEX++];
    }

    //複数の配列を受け取る
    public static int[] niahd(int n, int k) throws Exception {
        if (S_ARRAY == null) {
            S_ARRAY = new int[k][n];
            for (int j = 0; j < n; j++) {
                for (int i = 0; i < k; i++) {
                    S_ARRAY[i][j] = ni();
                }
            }
        }
        for (int i = 0; i < N; i++)
            S_ARRAY[S_INDEX][i]--;
        return S_ARRAY[S_INDEX++];
    }

    public static long[] nlahd(int n, int k) throws Exception {
        if (S_LARRAY == null) {
            S_LARRAY = new long[k][n];
            for (int j = 0; j < n; j++) {
                for (int i = 0; i < k; i++) {
                    S_LARRAY[i][j] = nl() - 1;
                }
            }
        }
        return S_LARRAY[S_LINDEX++];
    }

    public static char[] nca() {
        char[] a = next().toCharArray();
        return a;
    }


    public static String[] nsa(int n) {
        String[] res = new String[n];
        for (int i = 0; i < n; i++) {
            res[i] = n();
        }
        return res;
    }

    //スペースが入っている場合
    public static char[][] ncts(int h, int w) {
        char[][] a = new char[h][w];
        for (int i = 0; i < h; i++) {
            a[i] = nextLine().replace(" ", "").toCharArray();
        }
        return a;
    }

    public static char[][] nct(int h, int w) {
        char[][] a = new char[h][w];
        for (int hi = 0; hi < h; hi++) {
            String s = nextLine();
            for (int wi = 0; wi < s.length(); wi++) {
                a[hi][wi] = s.charAt(wi);
            }
        }
        return a;
    }

    public static char[][] nctp(int h, int w, char c) {
        char[][] a = new char[h + 2][w + 2];
        for (int hi = 1; hi < h + 1; hi++) {
            String s = nextLine();
            for (int wi = 1; wi < s.length() + 1; wi++) {
                a[hi][wi] = s.charAt(wi - 1);
            }
        }
        for (int wi = 0; wi < w + 2; wi++)
            a[0][wi] = a[h + 1][wi] = c;
        for (int hi = 0; hi < h + 2; hi++)
            a[hi][0] = a[hi][w + 1] = c;
        return a;
    }

    //スペースが入ってる時用
    public static char[][] nctsp(int h, int w, char c) {
        char[][] a = new char[h + 2][w + 2];
        //char c = '*';
        int i;
        for (i = 0; i < w + 2; i++)
            a[0][i] = c;
        for (i = 1; i < h + 1; i++) {
            a[i] = (c + nextLine().replace(" ", "") + c).toCharArray();
        }
        for (i = 0; i < w + 2; i++)
            a[h + 1][i] = c;
        return a;
    }

    public static long[] nla(int n) {
        long[] a = new long[n];
        for (int i = 0; i < n; i++) {
            a[i] = nl();
        }
        return a;
    }

    public static long[] nlas(int n, long e) {
        long[] a = new long[n + 1];
        for (int i = 0; i < n; i++) {
            a[i] = nl();
        }
        a[n] = e;
        return a;
    }

    public static long[] nlao(int n) {
        long[] a = new long[n + 1];
        for (int i = 0; i < n; i++) {
            a[i + 1] = nl();
        }
        return a;
    }

    public static long[] nlad(int n) {
        long[] a = new long[n];
        for (int i = 0; i < n; i++) {
            a[i] = nl() - 1;
        }
        return a;
    }

    public static long[][] nlt(int h, int w) {
        long[][] a = new long[h][w];
        for (int hi = 0; hi < h; hi++) {
            for (int wi = 0; wi < w; wi++) {
                a[hi][wi] = nl();
            }
        }
        return a;
    }

    //便利クラス
    static class CouMap {
        public HashMap<Long, Long> map;
        public HashMap<String, Long> smap;

        CouMap() {
            map = new HashMap();
            smap = new HashMap();
        }

        public void put(long key, long value) {
            Long nowValue = map.get(key);
            map.put(key, nowValue == null ? value : nowValue + value);
        }

        public void put(String key, long value) {
            Long nowValue = smap.get(key);
            smap.put(key, nowValue == null ? value : nowValue + value);
        }

        public void mput(long key, long value) {
            Long nowValue = map.get(key);
            map.put(key, nowValue == null ? value % MOD : mSum(nowValue, value));
        }

        public void put(long key) {
            put(key, 1);
        }

        public void put(String key) {
            put(key, 1);
        }

        public void put(int... arg) {
            for (int i : arg) {
                put(i, 1);
            }
        }

        public void put(long... arg) {
            for (long i : arg) {
                put(i, 1);
            }
        }

        public void mput(int... arg) {
            for (int i : arg) {
                mput(i, 1);
            }
        }

        public void mput(long... arg) {
            for (long i : arg) {
                mput(i, 1);
            }
        }

        public long get(long key) {
            Long v = map.get(key);
            return v == null ? 0 : v;
        }

        public long get(String key) {
            Long v = map.get(key);
            return v == null ? 0 : v;
        }
    }

    static class P implements Comparable<P> {
        int x, y;

        @Override
        public int compareTo(P p) {
            //xyで昇順
            return x == p.x ? y - p.y : x - p.x;
            //xyで降順
            //return (x == p.x ? y - p.y : x - p.x) * -1;
            //yxで昇順
            //return y == p.y ? x - p.x : y - p.y;
            //yxで昇順
            //return (y == p.y ? x - p.x : y - p.y) * -1;

            //x昇 y降
            //return x == p.x ? p.y - y : x - p.x;
            //x降 y昇
            //return (x == p.x ? p.y - y : x - p.x) * -1;
            //y昇 x降
            //return y == p.y ? p.x - x : y - p.y;
            //y降 x昇
            //return (y == p.y ? p.x - x : y - p.y) * -1;
        }

        P(int a, int b) {
            x = a;
            y = b;
        }

        @Override
        public boolean equals(Object o) {
            if (this == o) return true;
            if (!(o instanceof P)) return false;
            P p = (P) o;
            return x == p.x && y == p.y;
        }

        @Override
        public int hashCode() {
            return Objects.hash(x, y);
        }

    }

    static class PL implements Comparable<PL> {
        long x, y;

        public int compareTo(PL p) {
            //xyで昇順
            long res = x == p.x ? y - p.y : x - p.x;
            //xyで降順
            //long res = (x == p.x ? y - p.y : x - p.x) * -1;
            //yxで昇順
            //long res = y == p.y ? x - p.x : y - p.y;
            //yxで昇順
            //long res = (y == p.y ? x - p.x : y - p.y) * -1;

            //x昇 y降
            //long res = x == p.x ? p.y - y : x - p.x;
            //x降 y昇
            //long res = (x == p.x ? p.y - y : x - p.x) * -1;
            //y昇 x降
            //long res = y == p.y ? p.x - x : y - p.y;
            //y降 x昇
            //long res = (y == p.y ? p.x - x : y - p.y) * -1;

            return (res == 0) ? 0 : res > 0 ? 1 : -1;
        }

        PL(long a, long b) {
            x = a;
            y = b;
        }

        @Override
        public boolean equals(Object o) {
            if (this == o) return true;
            if (!(o instanceof PL)) return false;
            PL p = (PL) o;
            return x == p.x && y == p.y;
        }

        @Override
        public int hashCode() {
            return Objects.hash(x, y);
        }

    }

    //値を渡す際は半開区間
    static class RectangleSum {
        //半開区間 0は0
        long[][] rui;
        int H, W;

        RectangleSum(long[][] ori) {
            H = ori.length;
            W = ori[0].length;
            rui = new long[H + 1][W + 1];
            for (int hi = 0; hi < H; hi++) {
                for (int wi = 0; wi < W; wi++) {
                    rui[hi + 1][wi + 1] = ori[hi][wi];
                }
            }
            for (int hi = 1; hi < H + 1; hi++) {
                for (int wi = 1; wi < W + 1; wi++) {
                    rui[hi][wi] += rui[hi - 1][wi];
                    rui[hi][wi] += rui[hi][wi - 1];
                    rui[hi][wi] -= rui[hi - 1][wi - 1];
                }
            }
        }

        RectangleSum(int[][] ori) {
            H = ori.length;
            W = ori[0].length;
            rui = new long[H + 1][W + 1];
            for (int hi = 0; hi < H; hi++) {
                for (int wi = 0; wi < W; wi++) {
                    rui[hi + 1][wi + 1] = ori[hi][wi];
                }
            }
            for (int hi = 1; hi < H + 1; hi++) {
                for (int wi = 1; wi < W + 1; wi++) {
                    rui[hi][wi] += rui[hi - 1][wi];
                    rui[hi][wi] += rui[hi][wi - 1];
                    rui[hi][wi] -= rui[hi - 1][wi - 1];
                }
            }
        }

        //半開区間
        public long getSum(int left, int right, int top, int bottom) {
            if (right > W || bottom > H) return 0;
            if (left < 0 || top < 0) return 0;
            if (top >= bottom || left >= right) return 0;
            long res = rui[bottom][right];
            res -= rui[top][right];
            res -= rui[bottom][left];
            res += rui[top][left];
            return res;
        }

    }

    public static void main(String[] args) throws Exception {
        long startTime = System.currentTimeMillis();
        solve();
        System.out.flush();
        long endTime = System.currentTimeMillis();
        if (DEBUG) System.err.println(endTime - startTime);
    }

}