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 <bits/stdc++.h>

using namespace std;

using int64 = long long;
const int mod = 1e9 + 7;
// const int mod = 998244353;

const int64 infll = (1LL << 62) - 1;
const int inf = (1 << 30) - 1;

struct IoSetup {
  IoSetup() {
    cin.tie(nullptr);
    ios::sync_with_stdio(false);
    cout << fixed << setprecision(10);
    cerr << fixed << setprecision(10);
  }
} iosetup;


template< typename T1, typename T2 >
ostream &operator<<(ostream &os, const pair< T1, T2 > &p) {
  os << p.first << " " << p.second;
  return os;
}

template< typename T1, typename T2 >
istream &operator>>(istream &is, pair< T1, T2 > &p) {
  is >> p.first >> p.second;
  return is;
}

template< typename T >
ostream &operator<<(ostream &os, const vector< T > &v) {
  for(int i = 0; i < (int) v.size(); i++) {
    os << v[i] << (i + 1 != v.size() ? " " : "");
  }
  return os;
}

template< typename T >
istream &operator>>(istream &is, vector< T > &v) {
  for(T &in : v) is >> in;
  return is;
}

template< typename T1, typename T2 >
inline bool chmax(T1 &a, T2 b) { return a < b && (a = b, true); }

template< typename T1, typename T2 >
inline bool chmin(T1 &a, T2 b) { return a > b && (a = b, true); }

template< typename T = int64 >
vector< T > make_v(size_t a) {
  return vector< T >(a);
}

template< typename T, typename... Ts >
auto make_v(size_t a, Ts... ts) {
  return vector< decltype(make_v< T >(ts...)) >(a, make_v< T >(ts...));
}

template< typename T, typename V >
typename enable_if< is_class< T >::value == 0 >::type fill_v(T &t, const V &v) {
  t = v;
}

template< typename T, typename V >
typename enable_if< is_class< T >::value != 0 >::type fill_v(T &t, const V &v) {
  for(auto &e : t) fill_v(e, v);
}

template< typename F >
struct FixPoint : F {
  FixPoint(F &&f) : F(forward< F >(f)) {}

  template< typename... Args >
  decltype(auto) operator()(Args &&... args) const {
    return F::operator()(*this, forward< Args >(args)...);
  }
};

template< typename F >
inline decltype(auto) MFP(F &&f) {
  return FixPoint< F >{forward< F >(f)};
}

int main() {
  int N;
  cin >> N;
  vector< int64 > A(N);
  cin >> A;
  int64 nim = 0;
  for(int i = 2; i < N; i++) nim ^= A[i];

  auto dp = make_v< int64 >(52, 2, 2);
  fill_v(dp, -1);
  auto rec = MFP([&](auto rec, int bit, bool borrow, bool add) -> int64 {
    if(bit == 50) {
      if(borrow == 0 && add == 0) return 0;
      return infll;
    }
    if(~dp[bit][borrow][add]) {
      return dp[bit][borrow][add];
    }
    int x = (A[0] >> bit) & 1;
    int y = (A[1] >> bit) & 1;
    int z = (nim >> bit) & 1;
    x -= borrow;
    y += add;

    int64 ret = infll;

    if(x == -1) {
      if(y == 0) {
        // x:-1, y:0
        // x:-2, y:1
        if(z == 1) {
          chmin(ret, rec(bit + 1, true, false));
        }
      } else if(y == 1) {
        // x:-1, y:1
        // x:-2, y:2
        if(z == 0) {
          chmin(ret, rec(bit + 1, true, false));
          chmin(ret, rec(bit + 1, true, true) + (1LL << bit));
        }
      } else {
        // x:-1, y:2
        // x:-2, y:3
        if(z == 1) {
          chmin(ret, rec(bit + 1, true, true));
        }
      }
    } else if(x == 0) {
      if(y == 0) {
        // x:0, y:0
        // x:-1, y:1
        if(z == 0) {
          chmin(ret, rec(bit + 1, false, false));
          chmin(ret, rec(bit + 1, true, true) + (1LL << bit));
        }
      } else if(y == 1) {
        // x:0, y:1
        // x:-1, y:2
        if(z == 1) {
          chmin(ret, rec(bit + 1, false, false));
          chmin(ret, rec(bit + 1, true, true) + (1LL << bit));
        }
      } else {
        // x:0, y:2
        // x:-1,y:3
        if(z == 0) {
          chmin(ret, rec(bit + 1, false, true));
          chmin(ret, rec(bit + 1, true, true) + (1LL << bit));
        }
      }
    } else {
      if(y == 0) {
        // x:1, y:0
        // x:0, y:1
        if(z == 1) {
          chmin(ret, rec(bit + 1, false, false));
        }
      } else if(y == 1) {
        // x:1, y:1
        // x:0, y:2
        if(z == 0) {
          chmin(ret, rec(bit + 1, false, false));
          chmin(ret, rec(bit + 1, false, true) + (1LL << bit));
        }
      } else {
        // x:1, y:2
        // x:0, y:3
        if(z == 1) {
          chmin(ret, rec(bit + 1, false, true));
        }
      }
    }
    return dp[bit][borrow][add] = ret;
  });

  auto res = rec(0, false, false);
  if(res >= A[0]) res = -1;
  cout << res << "\n";
}