Problem:
Score : 700 points

Problem Statement

You are given N integers A_1, A_2, ..., A_N.

Consider the sums of all non-empty subsequences of A. There are 2^N - 1 such sums, an odd number.

Let the list of these sums in non-decreasing order be S_1, S_2, ..., S_{2^N - 1}.

Find the median of this list, S_{2^{N-1}}.

Constraints

1 \leq N \leq 2000

1 \leq A_i \leq 2000

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the median of the sorted list of the sums of all non-empty subsequences of A.

Sample Input 1

3
1 2 1

Sample Output 1

2

In this case, S = (1, 1, 2, 2, 3, 3, 4). Its median is S_4 = 2.

Sample Input 2

1
58

Sample Output 2

58

In this case, S = (58).

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
#include <unistd.h>

using namespace std;

#define DEBUG(x) cerr<<#x<<": "<<x<<endl;
#define DEBUG_VEC(v) cerr<<#v<<":";for(int i=0;i<v.size();i++) cerr<<" "<<v[i]; cerr<<endl
#define DEBUG_MAT(v) cerr<<#v<<endl;for(int i=0;i<v.size();i++){for(int j=0;j<v[i].size();j++) {cerr<<v[i][j]<<" ";}cerr<<endl;}


typedef long long ll;
#define vi vector<int>
#define vl vector<ll>
#define vii vector< vector<int> >
#define vll vector< vector<ll> >
#define vs vector<string>
#define pii pair<int,int>
#define pis pair<int,string>
#define psi pair<string,int>
//#define pll pair<ll,ll>
#define pll pair<unsigned long long, unsigned long long>
template<class S, class T> pair<S, T> operator+(const pair<S, T> &s, const pair<S, T> &t) { return pair<S, T>(s.first + t.first, s.second + t.second); }
template<class S, class T> pair<S, T> operator-(const pair<S, T> &s, const pair<S, T> &t) { return pair<S, T>(s.first - t.first, s.second - t.second); }
template<class S, class T> ostream& operator<<(ostream& os, pair<S, T> p) { os << "(" << p.first << ", " << p.second << ")"; return os; }
#define X first
#define Y second
#define rep(i,n) for(int i=0;i<(n);i++)
#define rep1(i,n) for(int i=1;i<=(n);i++)
#define rrep(i,n) for(int i=(n)-1;i>=0;i--)
#define rrep1(i,n) for(int i=(n);i>0;i--)
#define REP(i,a,b) for(int i=a;i<b;i++)
#define in(x, a, b) (a <= x && x < b)
#define all(c) c.begin(),c.end()
template<class T> bool chmax(T &a, const T &b) { if (a<b) { a = b; return 1; } return 0; }
template<class T> bool chmin(T &a, const T &b) { if (a>b) { a = b; return 1; } return 0; }
#define UNIQUE(v) v.erase(std::unique(v.begin(), v.end()), v.end());
const ll inf = 1000000001;
const ll INF = (ll)1e18 + 1;
const ll MOD = 1000000007;
//const ll MOD = 998244353;
const double pi = 3.14159265358979323846;
#define Sp(p) cout<<setprecision(15)<< fixed<<p<<endl;
int dx[4] = { 1,0, -1,0 }, dy[4] = { 0,1,0,-1 };
int dx2[8] = { 1,1,0,-1,-1,-1,0,1 }, dy2[8] = { 0,1,1,1,0,-1,-1,-1 };
#define fio() cin.tie(0); ios::sync_with_stdio(false);
//#define mp make_pair



typedef complex<double> P;
 
vector<P> fft(vector<P> v, bool rev = false) {
	int n = v.size(), i, j, m;
 
	for (i = 0, j = 1; j<n - 1; j++) {
		for (int k = n >> 1; k>(i ^= k); k >>= 1);
		if (i>j) swap(v[i], v[j]);
	}
	for (int m = 2; m <= n; m *= 2) {
		double deg = (rev ? -1 : 1) * 2 * acos(-1) / m;
		P wr(cos(deg), sin(deg));
		for (i = 0; i<n; i += m) {
			P w(1, 0);
			for (int j1 = i, j2 = i + m / 2; j2<i + m; j1++, j2++) {
				P t1 = v[j1], t2 = w*v[j2];
				v[j1] = t1 + t2, v[j2] = t1 - t2;
				w *= wr;
			}
		}
	}
	if (rev) rep(i, n) v[i] *= 1.0 / n;
	return v;
}
 
vector<P> MultPoly(vector<P> a, vector<P> b) {
  int n = a.size();
  while (a.size() < 2*n) {
    a.push_back(P(0, 0));
    b.push_back(P(0, 0));
  }
  a = fft(a), b = fft(b);
  for (int i = 0; i<a.size(); i++) a[i] *= b[i];
  return fft(a, true);
}

vi a;


vector<P> solve(int l, int r) {
  //DEBUG(pii(l, r));
  if (r - l == 1) {
    vector<P> res(a[l] + 1);
    res[0].real(1);
    res[a[l]].real(1);
    //DEBUG_VEC(res);
    return res;
  }
  auto lv = solve(l, (l + r) / 2), rv = solve((l + r) / 2, r);
  while (abs(lv.back().real()) <= 0.0000000001) {
    lv.pop_back();
  }
  while (abs(rv.back().real()) <= 0.0000000001) {
    rv.pop_back();
  }
  ll x = max(lv.size(), rv.size());
  while (x & (x - 1)) x++;
  x *= 4;
  while (lv.size() < x) {
    lv.push_back(P(0, 0));
  }
  while (rv.size() < x) {
    rv.push_back(P(0, 0));
  }
  //DEBUG_VEC(lv);
  //DEBUG_VEC(rv);
  auto res = MultPoly(lv, rv);
  rep (i, res.size()) {
    if (abs(res[i].real()) >= 0.000000001) {
      res[i].real(1);
    }
    else {
      res[i].real(0);
    }
    res[i].imag(0);
  }
  //DEBUG_VEC(res);
  //DEBUG_VEC(res);
  return res;
}


int main() {
  int n;
  cin >> n;
  a.resize(n);
  int sum = 0;
  rep (i, n) {
    cin >> a[i];
    sum += a[i];
  }
  if (n == 1) {
    cout << sum << endl;
    return 0;
  }
  auto ans = solve(0, n);
  for (int i = sum / 2; i > 0; i--) {
    if (ans[i].real() != 0) {
      cout << i << endl;
      return 0;
    }
  }
  
}