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)

//#pragma GCC optimize("Ofast")
//#pragma GCC target("avx")
//#undef LOCAL




#include <algorithm>

#include <array>

#include <bitset>

#include <cassert>

#include <complex>

#include <cstdio>

#include <cstring>

#include <iostream>

#include <map>

#include <numeric>

#include <queue>

#include <set>

#include <string>

#include <unordered_map>

#include <unordered_set>

#include <vector>

using namespace std;

using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n - 1); }
template <class T> using V = vector<T>;
template <class T> using VV = V<V<T>>;



#include <unistd.h>

struct Scanner {
    int fd = -1;
    char line[(1 << 15) + 1];
    size_t st = 0, ed = 0;
    void reread() {
        memmove(line, line + st, ed - st);
        ed -= st;
        st = 0;
        ed += ::read(fd, line + ed, (1 << 15) - ed);
        line[ed] = '\0';
    }
    bool succ() {
        while (true) {
            if (st == ed) {
                reread();
                if (st == ed) return false;
            }
            while (st != ed && isspace(line[st])) st++;
            if (st != ed) break;
        }
        if (ed - st <= 50) {
            bool sep = false;
            for (size_t i = st; i < ed; i++) {
                if (isspace(line[i])) {
                    sep = true;
                    break;
                }
            }
            if (!sep) reread();
        }
        return true;
    }
    template <class T, enable_if_t<is_same<T, string>::value, int> = 0>
    bool read_single(T& ref) {
        if (!succ()) return false;
        while (true) {
            size_t sz = 0;
            while (st + sz < ed && !isspace(line[st + sz])) sz++;
            ref.append(line + st, sz);
            st += sz;
            if (!sz || st != ed) break;
            reread();
        }
        return true;
    }
    template <class T, enable_if_t<is_integral<T>::value, int> = 0>
    bool read_single(T& ref) {
        if (!succ()) return false;
        bool neg = false;
        if (line[st] == '-') {
            neg = true;
            st++;
        }
        ref = T(0);
        while (isdigit(line[st])) {
            ref = 10 * ref + (line[st++] & 0xf);
        }
        if (neg) ref = -ref;
        return true;
    }
    template <class T> bool read_single(V<T>& ref) {
        for (auto& d : ref) {
            if (!read_single(d)) return false;
        }
        return true;
    }
    void read() {}
    template <class H, class... T> void read(H& h, T&... t) {
        bool f = read_single(h);
        assert(f);
        read(t...);
    }
    Scanner(FILE* fp) : fd(fileno(fp)) {}
};

struct Printer {
  public:
    template <bool F = false> void write() {}
    template <bool F = false, class H, class... T>
    void write(const H& h, const T&... t) {
        if (F) write_single(' ');
        write_single(h);
        write<true>(t...);
    }
    template <class... T> void writeln(const T&... t) {
        write(t...);
        write_single('\n');
    }

    Printer(FILE* _fp) : fp(_fp) {}
    ~Printer() { flush(); }

  private:
    static constexpr size_t SIZE = 1 << 15;
    FILE* fp;
    char line[SIZE], small[50];
    size_t pos = 0;
    void flush() {
        fwrite(line, 1, pos, fp);
        pos = 0;
    }
    void write_single(const char& val) {
        if (pos == SIZE) flush();
        line[pos++] = val;
    }
    template <class T, enable_if_t<is_integral<T>::value, int> = 0>
    void write_single(T val) {
        if (pos > (1 << 15) - 50) flush();
        if (val == 0) {
            write_single('0');
            return;
        }
        if (val < 0) {
            write_single('-');
            val = -val; // todo min
        }
        size_t len = 0;
        while (val) {
            small[len++] = char(0x30 | (val % 10));
            val /= 10;
        }
        for (size_t i = 0; i < len; i++) {
            line[pos + i] = small[len - 1 - i];
        }
        pos += len;
    }
    void write_single(const string& s) {
        for (char c : s) write_single(c);
    }
    void write_single(const char* s) {
        size_t len = strlen(s);
        for (size_t i = 0; i < len; i++) write_single(s[i]);
    }
    template <class T> void write_single(const V<T>& val) {
        auto n = val.size();
        for (size_t i = 0; i < n; i++) {
            if (i) write_single(' ');
            write_single(val[i]);
        }
    }
};

Scanner sc = Scanner(stdin);
Printer pr = Printer(stdout);

int main() {
    int n;
    sc.read(n);

    V<ll> a(n + 1);
    for (int i = 0; i <= n; i++) {
        sc.read(a[i]);
    }

    if (n == 0) {
        if (a[0] == 1) {
            pr.writeln(1);
        } else {
            pr.writeln(-1);
        }
        return 0;
    }

    if (a[0] != 0) {
        pr.writeln(-1);
        return 0;
    }

    V<ll> up(n + 1);
    up[0] = 1;
    for (int i = 1; i <= n; i++) {
        up[i] = up[i - 1] * 2 - a[i];
        up[i] = min(up[i], TEN(18));
        if (up[i] < 0 || (i < n && up[i] <= 0)) {
            pr.writeln(-1);
            return 0;
        }
    }
           ;
    ll nw = 0;
    ll ans = 0;
    for (int i = n; i >= 0; i--) {
        nw = min(nw, up[i]);
        nw += a[i];
        ans += nw;
                  ;
    }
    pr.writeln(ans);
    return 0;
}