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++ (Clang 10.0.0) (judge verdict: Time Limit Exceeded)

//#include<bits/stdc++.h>
#include<iostream>
#include<algorithm>
#include<cmath>
#include<vector>
#include<queue>
#include<set>

/*
#include<boost/multiprecision/cpp_int.hpp>
#include <boost/multiprecision/cpp_dec_float.hpp>
#include <boost/rational.hpp>
namespace mp = boost::multiprecision;
using Real = mp::number<mp::cpp_dec_float<1024>>;
using Bint = mp::cpp_int;
*/

using namespace std;

#define int long long
#define REP(i,s,e) for((i)=(s);(i)<(e);(i)++)
#define RREP(i,s,e) for((i)=((s)-1);(i)>=(e);(i)--)
#define FOR(i,n) for((i)=(0);(i)<(n);(i)++)
#define RFOR(i,n) for((i)=((n)-1);(i)>=(0);(i)--)
#define MOD 1000000007
#define NMAX 10000001

signed main(){
    int N;
    int A1,A2;
    int xr;

    cin>>N;
    cin>>A1>>A2;
    xr=0;
    int ans=0;

    for(int i=0;i<N-2;i++){
        int Ain=0;
        cin>>Ain;
        xr=xr^Ain;
    }
    for(int i=60;i>=0;i--){
        int xri=(xr>>i)%2;
        int A1i=(A1>>i)%2;
        int A2i=(A2>>i)%2;
        //cout<<pow(2,i)<<","<<xri<<","<<A1i<<","<<A2i<<endl;
                    int n=1;
        if((xri^A1i^A2i)==1){
            if(A1i==1&&A2i==1){
                for(int j=i-1;j>=0;j--){
                    n+=((A1>>j)%2)*pow(2,j);
                    //cout<<n<<endl;
                }
            }
            else {
                for(int j=i-1;j>=0;j--){
                    n+=(1-((A2>>j)%2))*pow(2,j);
                }
            }
            A1-=n;
            A2+=n;
            ans+=n;
        }
        //cout<<A1<<","<<A2<<","<<xr<<endl;
        xri=(xr>>i)%2;
        A1i=(A1>>i)%2;
        A2i=(A2>>i)%2;
        if((xri^A1i^A2i)==1||A1<=0){
            A1+=n;
            A2-=n;
            ans-=n;
            int idx=0;
            while(1){
                ans--;
                A1-=n;
                A2+=n;
                ans+=n;
                if((xri^A1i^A2i)==1||A1<=0){
                    A1+=n;
                    A2-=n;
                    ans-=n;
                    continue;
                }
                if(idx>=100000000||ans<=0){
                    cout<<"-1"<<endl;
                    return 0;
                }
                break;
            }
        }
    }

    cout<<ans<<endl;


}
//011
//101