Revise paste Untitled

#include <iostream>
#include <bits/stdc++.h>
#include <fstream>
#include <string>
#include <set>
#include <vector>
#define MOD 1000000007
#define ll long long
using namespace std;

string add(string a, string b) {
    if(a.size() < b.size()) {
        swap(a, b);
    }
    int i = (int) a.size() - 1, j = (int) b.size() - 1;
    string res = "";
    int carry = 0;
    while(j >= 0) {
        int sum = (a[i] - '0') + (b[j] - '0') + carry;
        if(sum > 9) {
            sum -= 10;
            carry = 1;
        }
        else {
            carry = 0;
        }
        res += (sum + '0');
        i--;
        j--;
    }
    while(i >= 0) {
        int sum = (a[i] - '0') + carry;
        if(sum > 9) {
            sum -= 10;
            carry = 1;
        }
        else {
            carry = 0;
        }
        res += (sum + '0');
        i--;
    }
    if(carry == 1) {
        res += "1";
    }
    reverse(res.begin(), res.end());
    return res;
}
string multiply(string a, string b) {
    int sz_a = (int) a.size();
    int sz_b = (int) b.size();
    vector<int> v(sz_a + sz_b, 0);
    
    for(int i = sz_a - 1; i >= 0; i--){
        for(int j = sz_b - 1; j >= 0; j--) {
            v[i + j + 1] += (a[i] - '0') * (b[j] - '0');
        }
    }
    for(int i = sz_a + sz_b - 1; i >= 0; i--) {
        if(v[i] > 9) {
            v[i - 1] += v[i] / 10;
            v[i] %= 10;
        }
    }
    int i = 0;
    while(v[i] == 0) {
        i++;
    }
    string res = "";
    while(i < sz_a + sz_b) {
        res += (v[i] + '0');
        i++;
    }
    if(res == "") {
        res = "0";
    }
    return res;
}
int main() {
    int n;
    cin >> n;
    
    string sum = "0";
    string power_of_two = "1";
    
    for(int i = 0; i < n; i++) {
        power_of_two = multiply(power_of_two, "2");
        sum = add(sum, power_of_two);
    }
    cout << sum << endl;
    return 0;
}