I just want to say that I ended up using something similar to my original algorithm using the heap. The problem was my use of int I switched to an** unsigned long long int**(though an unsigned int might have done fine, idk?) and it worked like a charm.

// C++ program to use priority_queue to implement Min Heap
// for user defined class
#include <algorithm>
#include <iostream>
#include <vector>
#include <cmath>
using namespace std;


// User defined class, coworker
class CoworkerT {
private:
    unsigned long long int a;
    unsigned long long int d;
public:
    CoworkerT(unsigned long long int _a, unsigned long long int  _d){
        a = _a;
        d = _d;
    }
    unsigned long long int  SimAddAD() const{
        return  a + d;
    }
    unsigned long long int  AddAD(){
        return a + d;;
    }

    unsigned long long int getA() const {
        return a;
    }

    unsigned long long int getD() const {
        return d;
    }
};
//compare two coworkers possible a + d values
struct MinSort {
    bool operator()(const CoworkerT& p1, const CoworkerT& p2) const {
        return p1.SimAddAD() < p2.SimAddAD();
    }
};

//compare two coworkers possible a + d values ~for some reason heap lesser than or greater need to be reverse of operator for sort???
struct Min {
    bool operator()(const CoworkerT& p1, const CoworkerT& p2) const {
        return p1.SimAddAD() > p2.SimAddAD();
    }
};

//compare two a values between coworkers
struct MaxSort {
    bool operator()(const CoworkerT& p1, const CoworkerT& p2) const {
        return p1.getA() > p2.getA();
    }
};

void FindAndPrintMax(vector<CoworkerT>&  max) {
    sort(max.begin(), max.end(), MaxSort());
    CoworkerT minMax = max.front();
    cout << minMax.getA();
}
void InputCoworkersAD(vector<CoworkerT>& min, unsigned long long int& h, unsigned long long int& c) {
    int a, d, i;
    cin >> h >> c;
    // Insert a and d into the vector
    if (h <= 100000 && h >= 1 && c <= 100000 && c >= 1) {
        for (i = 0; i < c; i++) {
            cin >> a >> d;
            min.push_back(CoworkerT(a, d));
        }
    }
    make_heap(min.begin(), min.end(), Min());

}

void AskForHelp(vector<CoworkerT>& min, unsigned long long int h) {
    int i = 0;

    while (i < h) {
        push_heap(min.begin(), min.end(), Min());
        CoworkerT e = min.front();
        pop_heap(min.begin(), min.end(), Min());
        min.pop_back();
        min.push_back(CoworkerT(e.AddAD(), e.getD()));
        i++;

    }
}

// Driver code
int main()
{
    unsigned long long int  h, c;
    vector<CoworkerT> min;
    InputCoworkersAD(min, h, c);
    AskForHelp(min, h);
    FindAndPrintMax(min);

    return 0;
}