Problem:
Score : 600 points

Problem Statement

There are N piles of stones. The i-th pile has A_i stones.

Aoki and Takahashi are about to use them to play the following game:

Starting with Aoki, the two players alternately do the following operation:

Operation: Choose one pile of stones, and remove one or more stones from it.

When a player is unable to do the operation, he loses, and the other player wins.

When the two players play optimally, there are two possibilities in this game: the player who moves first always wins, or the player who moves second always wins, only depending on the initial number of stones in each pile.

In such a situation, Takahashi, the second player to act, is trying to guarantee his win by moving at least zero and at most (A_1 - 1) stones from the 1-st pile to the 2-nd pile before the game begins.

If this is possible, print the minimum number of stones to move to guarantee his victory; otherwise, print -1 instead.

Constraints

2 \leq N \leq 300

1 \leq A_i \leq 10^{12}

Input

Input is given from Standard Input in the following format:

N
A_1 \ldots A_N

Output

Print the minimum number of stones to move to guarantee Takahashi's win; otherwise, print -1 instead.

Sample Input 1

2
5 3

Sample Output 1

1

Without moving stones, if Aoki first removes 2 stones from the 1-st pile, Takahashi cannot win in any way.

If Takahashi moves 1 stone from the 1-st pile to the 2-nd before the game begins so that both piles have 4 stones, Takahashi can always win by properly choosing his actions.

Sample Input 2

2
3 5

Sample Output 2

-1

It is not allowed to move stones from the 2-nd pile to the 1-st.

Sample Input 3

3
1 1 2

Sample Output 3

-1

It is not allowed to move all stones from the 1-st pile.

Sample Input 4

8
10 9 8 7 6 5 4 3

Sample Output 4

3

Sample Input 5

3
4294967297 8589934593 12884901890

Sample Output 5

1

Watch out for overflows.

Solution in C++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <array>
#include <vector>
#include <map>
#include <unordered_map>
#include <set>
#include <unordered_set>
#include <algorithm>
#include <math.h>
#include <string>
#include <climits>
#include <assert.h>
#include <iomanip>
#include <bitset>
#include <queue>
#include <deque>
#include <stack>
#include <functional>
#include <fstream>
#include <random>

#define LEN(x) (long long)(x.size())
#define FOR(i, a, n) for(int i=(a);i<(n); ++i)
#define FOE(i, a) for(auto i : a)
#define ALL(c) (c).begin(), (c).end()
#define RALL(c) (c).rbegin(), (c).rend()
#define SUM(x) std::accumulate(ALL(x), 0LL)
#define MIN(v) *std::min_element(v.begin(), v.end())
#define MAX(v) *std::max_element(v.begin(), v.end())
#define EXIST(v, x) (std::find(v.begin(), v.end(), x) != v.end())
#define BIT_COUNT(bit) (__builtin_popcount(bit))

typedef long long LL;
template<typename T> std::vector<T> make_v(size_t a){return std::vector<T>(a);}
template<typename T,typename... Ts> auto make_v(size_t a, Ts... ts){ return std::vector<decltype(make_v<T>(ts...))>(a,make_v<T>(ts...));}    // C++14
template<typename T,typename V> typename std::enable_if<std::is_class<T>::value==0>::type fill_v(T &t,const V &v){t=v;}
template<typename T,typename V> typename std::enable_if<std::is_class<T>::value!=0>::type fill_v(T &t,const V &v){for(auto &e:t) fill_v(e,v);}
template<class T> inline T ceil(T a, T b) { return (a + b - 1) / b; }
void print() { std::cout << std::endl; }
template <class Head, class... Tail> void print(Head&& head, Tail&&... tail) { std::cout << head; if (sizeof...(tail) != 0) {std::cout << " ";} print(std::forward<Tail>(tail)...); }
template <class T> void print(std::vector<T> &v) {for (auto& a : v) { std::cout << a; if (&a != &v.back()) {std::cout << " ";} }std::cout << std::endl;}
template <class T> void print(std::vector<std::vector<T>> &vv) { for (auto& v : vv) { print(v); }}
void debug() { std::cerr << std::endl; }
template <class Head, class... Tail> void debug(Head&& head, Tail&&... tail) { std::cerr << head; if (sizeof...(tail) != 0) {std::cerr << " ";} print(std::forward<Tail>(tail)...); }
template <class T> void debug(std::vector<T> &v) {for (auto& a : v) { std::cerr << a; if (&a != &v.back()) {std::cerr << " ";} }std::cerr << std::endl;}
template <class T> void debug(std::vector<std::vector<T>> &vv) { for (auto& v : vv) { print(v); }}
inline bool inside(long long y, long long x, long long H, long long W) {return 0 <= y and y < H and 0 <= x and x < W; }
template<class T> inline double euclidean_distance(T y1, T x1, T y2, T x2) { return sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2)); }
template<class T> inline double manhattan_distance(T y1, T x1, T y2, T x2) { return abs(x1 - x2) + abs(y1 - y2); }
template<typename T> T &chmin(T &a, const T &b) { return a = std::min(a, b); }
template<typename T> T &chmax(T &a, const T &b) { return a = std::max(a, b); }
bool is_bit_on(const unsigned long long bit, const unsigned int i) { return (bit >> i) & 1u; }
unsigned long long bit_set(const unsigned long long bit, const unsigned int i, const unsigned int b) {
    assert(b == 0 or b == 1);
    if (b == 0) { return bit & ~(1ull << i); }
    else        {return bit | (1ull << i); }
}

// 初項s交差d長さnの数列の和
long long sum_of_arithmetic_progression(long long s, long long d, long long n) {
    return n * (2 * s + (n - 1) * d) / 2;
}

// xが2の階乗かどうか判定
bool is_power_of_two(long long x) {
    return !(x & (x - 1));
}

long long gcd(long long a, long long b) {
    if (b == 0) { return a; }
    return gcd(b, a % b);
}

long long gcd(std::vector<long long> &v) {
    long long ans = v[0];
    for (int i = 1; i < (int) v.size(); ++i) {
        ans = gcd(ans, v[i]);
    }
    return ans;
}

long long lcm(long long a, long long b) {
    long long g = gcd(a, b);
    return a / g * b;
}

const int INF = 1u << 30u;  // 1,073,741,824
const long long LINF = 1ull << 60u;
const double EPS = 1e-9;
const long double PI = acos(-1.0);
const std::vector<int> dy2 = {0, 1}, dx2 = {1, 0};
const std::vector<int> dy4 = {0, 1, 0, -1}, dx4 = {1, 0, -1, 0};
const std::vector<int> dy8 = {0, -1, 0, 1, 1, -1, -1, 1}, dx8 = {1, 0, -1, 0, 1, 1, -1, -1};

using namespace std;

int solve(vector<unsigned long long> &A) {
    unsigned long long x = 0;
    FOE(a, A) {
        x ^= a;
    }
    if (x == 0) {
        return 0;
    }

    x ^= A[0];
    x ^= A[1];

    const unsigned long long total = A[0] + A[1];
    const unsigned long long t = total - x;
    if (x > total or t % 2 != 0) {
        return -1;
    }
    unsigned long long a_and_b = t / 2;
    unsigned long long ans = a_and_b;
    for (int b = 50; b >= 0; --b) {
        if (is_bit_on(x, b)) {
            if (bit_set(ans, b, 1) <= A[0]) {
                ans = bit_set(ans, b, 1);
            }
        }
    }

    LL move = A[0] - ans;
    if (((A[0] - move) ^ (A[1] + move) ^ x) != 0) {
        return -1;
    }
    if (move < 0 or A[0] - move < 1) {
        return -1;
    }
    else {
        return move;
    }
}


int check(vector<unsigned long long> &A) {
    unsigned long long entire_xor = 0;
    FOE(a, A) {
        entire_xor ^= a;
    }
    if (entire_xor == 0) {
        return 0;
    }

    entire_xor ^= A[0];
    entire_xor ^= A[1];

    LL total = A[0] + A[1];

    FOR(i, 0, A[0]) {
        if (((A[0] - i) ^ (A[1] + i) ^ entire_xor) == 0) {
            return i;
        }
    }

    return -1;
}


int main() {
    cin.tie(0);
    ios::sync_with_stdio(false);

    LL N;
    cin >> N;
    auto A = make_v<unsigned long long>(N);
    FOR(i, 0, N) {
        cin >> A[i];
    }

    print(solve(A));

    return 0;
}