Problem:
Score : 600 points

Problem Statement

Given is an integer sequence of length N+1: A_0, A_1, A_2, \ldots, A_N. Is there a binary tree of depth N such that, for each d = 0, 1, \ldots, N, there are exactly A_d leaves at depth d? If such a tree exists, print the maximum possible number of vertices in such a tree; otherwise, print -1.

Notes

A binary tree is a rooted tree such that each vertex has at most two direct children.

A leaf in a binary tree is a vertex with zero children.

The depth of a vertex v in a binary tree is the distance from the tree's root to v. (The root has the depth of 0.)

The depth of a binary tree is the maximum depth of a vertex in the tree.

Constraints

0 \leq N \leq 10^5

0 \leq A_i \leq 10^{8} (0 \leq i \leq N)

A_N \geq 1

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_0 A_1 A_2 \cdots A_N

Output

Print the answer as an integer.

Sample Input 1

3
0 1 1 2

Sample Output 1

7

Below is the tree with the maximum possible number of vertices. It has seven vertices, so we should print 7.

Sample Input 2

4
0 0 1 0 2

Sample Output 2

10

Sample Input 3

2
0 3 1

Sample Output 3

-1

Sample Input 4

1
1 1

Sample Output 4

-1

Sample Input 5

10
0 0 1 1 2 3 5 8 13 21 34

Sample Output 5

264

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <iostream>
#include <iomanip>
#include <cmath>
#include <algorithm>
#include <vector>
#include <set>
#include <unordered_set>
#include <map>
#include <unordered_map>
#include <random>
#include <chrono>
#include <string>
#include <cstring>
#include <bitset>
#include <cassert>
#include <queue>
#include <deque>
#include <memory>

//#pragma comment(linker, "/STACK:16777216")
//#pragma GCC optimize("O3")
//#pragma GCC optimize("Ofast")
//#pragma GCC optimize("unroll-loops")
//#pragma GCC optimize("unswitch-loops")
//#pragma GCC target("sse,sse2,sse3,ssse3,sse4,sse4.1,sse4.2,abm,mmx,avx,avx2,popcnt,tune=native")

using namespace std;

mt19937 rnd(chrono::steady_clock::now().time_since_epoch().count());

#define ll long long
#define ull unsigned long long
//#define double long double
#define int ll
//#define int ull
//#define int short
//#define int unsigned short
//#define int unsigned int
#define mp make_pair
#define F first
#define S second
#define pii pair<int, int>
#define pdd pair<double, double>
#define pid pair<int, double>
#define pdi pair<double, int>
#define hash hash228
const int SZ = 1e5 + 4;
const double EPS = 1e-7;
const int mod = 998244353;
const int hashpow = 2;
const int inf = 2e9;

struct custom_hash {
    static uint64_t splitmix64(uint64_t x) {
        // http://xorshift.di.unimi.it/splitmix64.c
        x += 0x9e3779b97f4a7c15;
        x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
        x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
        return x ^ (x >> 31);
    }

    size_t operator()(uint64_t x) const {
        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();
        return splitmix64(x + FIXED_RANDOM);
    }
};

int n;
int a[SZ];

inline void read() {
    cin >> n;
    for (int i = 0; i < n + 1; ++i) {
        cin >> a[i];
    }
}

inline void solve() {
    if (a[0]) {
        if (n == 0 && a[0] == 1) {
            cout << 1 << "\n";
        } else {
            cout << -1 << "\n";
        }
        return;
    }
    if (n == 0 && a[0] == 0) {
        cout << -1 << "\n";
        return;
    }
    vector<int> pref(n + 1, 0);
    for (int i = 1; i <= n; ++i) {
        pref[i] = pref[i - 1] + a[i];
    }
    int x = 1;
    int ans = 0;
    for (int i = 0; i <= n; ++i) {
        ans += x;
        x -= a[i];
        if (x < 0) {
            cout << -1 << "\n";
            return;
        }
        x = min(x * 2, pref[n] - pref[i]);
    }
    cout << ans << "\n";
}

signed main() {

#ifdef __LOCAL
    freopen("input.txt", "r", stdin);
    freopen("output.txt", "w", stdout);
#else
    ios_base::sync_with_stdio(0);
    cin.tie(0);
#endif

//    int t;
//    cin >> t;
//    for (int i = 0; i < t; ++i) {
    read();
    solve();
//    }

    return 0;
}