Language: C++

#include<bits/stdc++.h>
using namespace std;

#define FOR(i,bg,ed) for(int i=(bg);i<(ed);i++)
#define REP(i,n) FOR(i,0,n)
#define ALL(v) (v).begin(),(v).end()
#define fi first
#define se second
#define pb push_back

#define IL inline 
namespace geo {
    typedef double D;
    typedef bool B;
    typedef complex<D> P;
#define X real()
#define Y imag()
    B operator<(const P& l, const P& r){ return (l.X == r.X) ? l.Y < r.Y : l.X < r.X; }
    typedef pair<P, P> L;  //line
    typedef pair<P, P> LS; //line segment
    typedef pair<P, D> C;  //circle
    typedef vector<P> Poly;
    const D EPS = 1e-8;
    //Decompotision Macro
#define DCl(a,b,l) P (a),(b);tie(a,b)=l; 
#define DCc(a,b,c) P (a);D (b);tie(a,b)=c; 
    
    //A dot B
    IL D dot(P a, P b) { return a.X*b.X + a.Y*b.Y; }
    //A cross B
    IL D cross(P a, P b) { return a.X*b.Y - a.Y*b.X; }

    IL int sgn(P a,P b, P c){
        if(cross(b-a,c-a) > EPS)return 1;
        if(cross(b-a,c-a) < -EPS)return -1;
        if(dot(b-a,c-a) < -EPS)return 2; // b-a-c
        if(dot(a-b,c-b) < -EPS)return -2; // a-b-c
        return 0;
    }
    IL D signed_distance_P_L(P p, L l) {
        DCl(a,b,l);
        return cross(b - a, p - a) / abs(b - a);
    }
    
    IL D distance_P_L(P p, L l) {
        return abs(signed_distance_P_L(p,l));
    }

    IL D distance_P_LS(P p, LS ls) {
        DCl(a,b,ls);
        if (dot(b - a, p - a) < EPS) return abs(p - a);
        if (dot(a - b, p - b) < EPS) return abs(p - b);
        return abs(cross(b - a, p - a)) / abs(b - a);
    }

    
    IL B is_L_L(L p, L q) {
        DCl(a,b,p);
        DCl(c,d,q);
        return
            abs(cross(a - b, c - d)) > EPS ||
            abs(cross(a - b ,d - b)) < EPS;
    }
    IL B is_L_LS(L l, LS ls) {
        DCl(a,b,l);
        DCl(c,d,ls);
        return cross(a - b,c - b) * cross(a - b,d - b) <EPS;
    }
    IL B is_LS_LS(LS p, LS q) {
        DCl(a,b,p);
        DCl(c,d,q);
        return sgn(a,b,c)*sgn(a,b,d) <= 0
            && sgn(c,d,a)*sgn(c,d,b) <= 0;
    }

    IL D distance_LS_LS(LS p,LS q){
        if(is_LS_LS(p,q))return 0;
        return min({
                distance_P_LS(p.fi,q),
                distance_P_LS(p.se,q),
                distance_P_LS(q.fi,p),
                distance_P_LS(q.se,p)    
                    });
    }

    //caution: have to use isintersected_L_L before this function
    IL P cross_point_L_L(L p, L q) {
        DCl(a,b,p);
        D d1 = signed_distance_P_L(a, q);
        D d2 = signed_distance_P_L(b, q);
        return (a * d2 - b * d1) / (d2 - d1);
    }
    
    IL B is_C_C(C p, C q) {
        DCc(cp,rp,p);
        DCc(cq,rq,q);
        D d = abs(cp - cq);
        return abs(rp - rq) - d <EPS &&
                                 rp + rq - d > -EPS;
    }
    IL LS cross_C_C(C p, C q) {
        DCc(cp,rp,p);
        DCc(cq,rq,q);
        D d = abs(cp - cq);
        D rc = (d*d + rp*rp - rq*rq) / (2 * d);
        D rs = sqrt(rp*rp - rc*rc);
        P diff = (cq - cp) / d;
        return {cp + P(rc, rs)*diff, cp + P(rc, -rs)*diff};
    }
    IL D ccw(P a,P b,P c){
        return cross(b - a, c - a);
    }
    IL Poly convex_hull(Poly& p) {
        int n = (int)p.size();
        Poly res(2 * n);
        int k = 0, t;
        sort(ALL(p));
        REP(i,n) {
            while (k > 1 && ccw(res[k-2],res[k-1],p[i]) < EPS)k--;
            res[k++] = p[i];
        }
        t = k;
        REP(j,n-1){
            int i= n - 2 - j;
            while (k > t && ccw(res[k - 2],res[k - 1], p[i]) < EPS)k--;
            res[k++] = p[i];
        }
        res.resize(k - 1);
        return res;
    }
    IL B convex_in(Poly& l,P& p) {
        int a = 0, b = (int)l.size(), c;
        D A, C;
        P g = (l[a] + l[b - 1] + l[b / 2]) / 3.0;
        while (b - a > 1) {
            c = (a + b) / 2;
            A = cross(l[a] - g, p - l[a]);
            C = cross(l[c] - g, p - l[c]);
            if (cross(l[a] - g, l[c] - g) >= 0) {
                if (A > -EPS&&C < -EPS)b = c;
                else a = c;
            }
            else {
                if (C < -EPS || A > -EPS)b = c;
                else a = c;
            }
        }
        return(cross(l[b%l.size()] - l[a], p - l[b%l.size()]) > -EPS);
    }
    IL D convex_area(Poly& l) {
        double res = 0;
        int n = l.size();
        FOR(i,2,n)
            res += (abs(cross(l[i - 1] - l[0], l[i] - l[i - 1]))) / 2.0;
        return res;
    }
    inline vector<L> tangent_line(C c, P p) {
        vector<L> res;
        DCc(cp,r,c);
        P v = p - cp;
        D dist = abs(v);
        if (dist > r + EPS) {
            double _cos = r / dist;
            double _sin = sqrt(1.0 - _cos*_cos);
            v = v * (r / dist);
            {
                P e = P(_cos * v.X - _sin * v.Y, _sin*v.X + _cos*v.Y);
                res.push_back({p, cp + e});
            }
            {
                P e = P( _cos*v.X + _sin*v.Y ,-_sin*v.X + _cos*v.Y );
                res.push_back({p, cp + e});
            }
        }
        return res;
    }
    IL vector<L> inner_common_tangent_line(C p, C q) {
        vector<L> res;
        DCc(Mp,Rp,p);
        DCc(Mq,Rq,q);
        D dist = abs(Mp - Mq);
        if (dist > Rp + Rq + EPS) {
            P mid = (Mq*Rp + Mp*Rq) / (Rp + Rq);
            res = tangent_line(p, mid);
        }
        return res;
    }
    inline vector<L> outer_common_tangent_line(C p, C q) {
        vector<L> res;
        DCc(Mp,Rp,p);
        DCc(Mq,Rq,q);
        P v = Mp - Mq;
        D dist = abs(v);
        if (dist > Rp - Rq + EPS) {
            if (abs(Rp - Rq) < EPS) {
                P e = v*P( 0, 1 );
                e = e / dist;
                res.pb(L(Mp + e, Mq + e));
                res.pb(L(Mp - e, Mq - e));
            }
            else {
                P mid = Mp + (Mq - Mp) * Rp / (Rp - Rq);
                res = tangent_line(p, mid);
            }
        }
        return res;
    }
}

inline int check(geo::Line l,vector<geo::Circle>& cd, vector<geo::Circle>& cm){
    int res = 0;
    int n = (int)cd.size();
    for (int i = 0; i < n; i++){
        double d = geo::distance( cd[i].first,l);
    //  cout << d << " ";
        if (d<cm[i].second +geo::eps&&d>cd[i].second-geo::eps)
            res++;
    }
    //cout << endl;
    return res;
 
 
}
int main(){
    //ifstream cin("in.txt");ofstream cout("out.txt");
    while (1){
        int N;
        cin >> N;
        if (N == 0)break;
 
        vector<geo::Circle> cd(N);
        vector<geo::Circle> cm(N);
        for (int i = 0; i < N; i++){
            double x, y, d, m;
            cin >> x >> y >> d >> m;
            geo::x(cd[i].first, x);
            geo::y(cd[i].first, y);
            geo::x(cm[i].first, x);
            geo::y(cm[i].first, y);
            cd[i].second = d;
            cm[i].second = d+m;
        }
        if (N == 1){
            if (cm[0].second < geo::eps)cout << 0 << endl;
            else cout << 1 << endl;
            continue;
        }
        int res = 0;
        for (int i = 0; i < N; i++){
            for (int j = 0; j < N; j++){
                if (i != j){
                    //cout << i << j << endl << endl;
                    //cout << "inner1" << endl;
                    auto p1 = geo::inner_common_tangent_line(cd[i], cd[j]);
                    for (auto& it : p1){
                        res=max(check(it, cd, cm),res);
                    }
                //  cout << "inner2" << endl;
                    auto p2 = geo::inner_common_tangent_line(cd[i], cm[j]);
                    for (auto& it : p2){
                        res = max(check(it, cd, cm), res);
                    }
                //  cout << "inner3" << endl;
                    auto p3 = geo::inner_common_tangent_line(cm[i], cm[j]);
                    for (auto& it : p3){
                        res = max(check(it, cd, cm), res);
                    }
                //  cout << "outer1" << endl;
                    auto q1 = geo::outer_common_tangent_line(cd[i], cd[j]);
                    for (auto& it : q1){
                        res = max(check(it, cd, cm), res);
                    }
                //  cout << "outer2" << endl;
                    auto q2 = geo::outer_common_tangent_line(cd[i], cm[j]);
                    for (auto& it : q2){
                        res = max(check(it, cd, cm), res);
                    }
                //  cout << "outer3" << endl;
                    auto q3 = geo::outer_common_tangent_line(cm[i], cm[j]);
                    for (auto& it : q3){
                        res = max(check(it, cd, cm), res);
                    }
 
                }
            }
        }
        cout << res << endl;
    }
    //cin.close();  cout.close();
    return(0);
}

Compiler output:
a.cc:210:23: error: 'Line' is not a member of 'geo'
  210 | inline int check(geo::Line l,vector<geo::Circle>& cd, vector<geo::Circle>& cm){
      |                       ^~~~
a.cc:210:42: error: 'Circle' is not a member of 'geo'
  210 | inline int check(geo::Line l,vector<geo::Circle>& cd, vector<geo::Circle>& cm){
      |                                          ^~~~~~
a.cc:210:48: error: template argument 1 is invalid
  210 | inline int check(geo::Line l,vector<geo::Circle>& cd, vector<geo::Circle>& cm){
      |                                                ^
a.cc:210:48: error: template argument 2 is invalid
a.cc:210:51: error: 'cd' was not declared in this scope
  210 | inline int check(geo::Line l,vector<geo::Circle>& cd, vector<geo::Circle>& cm){
      |                                                   ^~
a.cc:210:67: error: 'Circle' is not a member of 'geo'
  210 | inline int check(geo::Line l,vector<geo::Circle>& cd, vector<geo::Circle>& cm){
      |                                                                   ^~~~~~
a.cc:210:73: error: template argument 1 is invalid
  210 | inline int check(geo::Line l,vector<geo::Circle>& cd, vector<geo::Circle>& cm){
      |                                                                         ^
a.cc:210:73: error: template argument 2 is invalid
a.cc:210:76: error: 'cm' was not declared in this scope; did you mean 'tm'?
  210 | inline int check(geo::Line l,vector<geo::Circle>& cd, vector<geo::Circle>& cm){
      |                                                                            ^~
      |                                                                            tm
a.cc:210:78: error: expression list treated as compound expression in initializer [-fpermissive]
  210 | inline int check(geo::Line l,vector<geo::Circle>& cd, vector<geo::Circle>& cm){
      |                                                                              ^
a.cc: In function 'int main()':
a.cc:231:21: error: 'Circle' is not a member of 'geo'
  231 |         vector<geo::Circle> cd(N);
      |                     ^~~~~~
a.cc:231:27: error: template argument 1 is invalid
  231 |         vector<geo::Circle> cd(N);
      |                           ^
a.cc:231:27: error: template argument 2 is invalid
a.cc:232:21: error: 'Circle' is not a member of 'geo'
  232 |         vector<geo::Circle> cm(N);
      |                     ^~~~~~
a.cc:232:27: error: template argument 1 is invalid
  232 |         vector<geo::Circle> cm(N);
      |                           ^
a.cc:232:27: error: template argument 2 is invalid
a.cc:236:18: error: 'x' is not a member of 'geo'
  236 |             geo::x(cd[i].first, x);
      |                  ^
a.cc:236:22: error: invalid types 'int[int]' for array subscript
  236 |             geo::x(cd[i].first, x);
      |                      ^
a.cc:237:18: error: 'y' is not a member of 'geo'
  237 |             geo::y(cd[i].first, y);
      |                  ^
a.cc:237:22: error: invalid types 'int[int]' for array subscript
  237 |             geo::y(cd[i].first, y);
      |                      ^
a.cc:238:18: error: 'x' is not a member of 'geo'
  238 |             geo::x(cm[i].first, x);
      |                  ^
a.cc:238:22: error: invalid types 'int[int]' for array subscript
  238 |             geo::x(cm[i].first, x);
      |                      ^
a.cc:239:18: error: 'y' is not a member of 'geo'
  239 |             geo::y(cm[i].first, y);
      |                  ^
a.cc:239:22: error: invalid types 'int[int]' for array subscript
  239 |             geo::y(cm[i].first, y);
      |                      ^
a.cc:240:15: error: invalid types 'int[int]' for array subscript
  240 |             cd[i].second = d;
      |               ^
a.cc:241:15: error: invalid types 'int[int]' for array subscript
  241 |             cm[i].second = d+m;
      |               ^
a.cc:244:19: error: invalid types 'int[int]' for array subscript
  244 |             if (cm[0].second < geo::eps)cout << 0 << endl;
      |                   ^
a.cc:244:37: error: 'eps' is not a member of 'geo'
  244 |             if (cm[0].second < geo::eps)cout << 0 << endl;
      |                                     ^~~
a.cc:254:64: error: invalid types 'int[int]' for array subscript
  254 |                     auto p1 = geo::inner_common_tangent_line(cd[i], cd[j]);
      |                                                                ^
a.cc:254:71: error: invalid types 'int[int]' for array subscript
  254 |                     auto p1 = geo::inner_common_tangent_line(cd[i], cd[j]);
      |                                                                       ^
a.cc:256:38: error: 'check' cannot be used as a function
  256 |                         res=max(check(it, cd, cm),res);
      |                                 ~~~~~^~~~~~~~~~~~
a.cc:259:64: error: invalid types 'int[int]' for array subscript
  259 |                     auto p2 = geo::inner_common_tangent_line(cd[i], cm[j]);
      |                                                                ^
a.cc:259:71: error: invalid types 'int[int]' for array subscript
  259 |                     auto p2 = geo::inner_common_tangent_line(cd[i], cm[j]);
      |                                                                       ^
a.cc:261:40: error: 'check' cannot be used as a function
  261 |                         res = max(check(it, cd, cm), res);
      |                                   ~~~~~^~~~~~~~~~~~
a.cc:264:64: error: invalid types 'int[int]' for array subscript
  264 |                     auto p3 = geo::inner_common_tangent_line(cm[i], cm[j]);
      |                                                                ^
a.cc:264:71: error: invalid types 'int[int]' for array subscript
  264 |                     auto p3 = geo::inner_common_tangent_line(cm[i], cm[j]);
      |                                                                       ^
a.cc:266:40: error: 'check' cannot be used as a function
  266 |                         res = max(check(it, cd, cm), res);
      |                                   ~~~~~^~~~~~~~~~~~
a.cc:269:64: error: invalid types 'int[int]' for array subscript
  269 |                     auto q1 = geo::outer_common_tangent_line(cd[i], cd[j]);
      |                                                                ^
a.cc:269:71: error: invalid types 'int[int]' for array subscript
  269 |                     auto q1 = geo::outer_common_tangent_line(cd[i], cd[j]);
      |                                                                       ^
a.cc:271:40: error: 'check' cannot be used as a function
  271 |                         res = max(check(it, cd, cm), res);
      |                                   ~~~~~^~~~~~~~~~~~
a.cc:274:64: error: invalid types 'int[int]' for array subscript
  274 |                     auto q2 = geo::outer_common_tangent_line(cd[i], cm[j]);
      |                                                                ^
a.cc:274:71: error: invalid types 'int[int]' for array subscript
  274 |                     auto q2 = geo::outer_common_tangent_line(cd[i], cm[j]);
      |                                                                       ^
a.cc:276:40: error: 'check' cannot be used as a function
  276 |                         res = max(check(it, cd, cm), res);
      |                                   ~~~~~^~~~~~~~~~~~
a.cc:279:64: error: invalid types 'int[int]' for array subscript
  279 |                     auto q3 = geo::outer_common_tangent_line(cm[i], cm[j]);
      |                                                                ^
a.cc:279:71: error: invalid types 'int[int]' for array subscript
  279 |                     auto q3 = geo::outer_common_tangent_line(cm[i], cm[j]);
      |                                                                       ^
a.cc:281:40: error: 'check' cannot be used as a function
  281 |                         res = max(check(it, cd, cm), res);
      |                                   ~~~~~^~~~~~~~~~~~
In file included from /usr/include/c++/14/bits/stl_algobase.h:71,
                 from /usr/include/c++/14/algorithm:60,
                 from /usr/include/x86_64-linux-gnu/c++/14/bits/stdc++.h:51,
                 from a.cc:1:
/usr/include/c++/14/bits/predefined_ops.h: In instantiation of 'constexpr bool __gnu_cxx::__ops::_Iter_less_iter::operator()(_Iterator1, _Iterator2) const [with _Iterator1 = __gnu_cxx::__normal_iterator<std::complex<double>*, std::vector<std::complex<double> > >; _Iterator2 = __gnu_cxx::__normal_iterator<std::complex<double>*, std::vector<std::complex<double> > >]':
/usr/include/c++/14/bits/stl_algo.h:1777:14:   required from 'void std::__insertion_sort(_RandomAccessIterator, _RandomAccessIterator, _Compare) [with _RandomAccessIterator = __gnu_cxx::__normal_iterator<complex<double>*, vector<complex<double> > >; _Compare = __gnu_cxx::__ops::_Iter_less_iter]'
 1777 |           if (__comp(__i, __first))
      |               ~~~~~~^~~~~~~~~~~~~~
/usr/include/c++/14/bits/stl_algo.h:1817:25:   required from 'void std::__final_insertion_sort(_RandomAccessIterator, _RandomAccessIterator, _Compare) [with _RandomAccessIterator = __gnu_cxx::__normal_iterator<complex<double>*, vector<complex<double> > >; _Compare = __gnu_cxx::__ops::_Iter_less_iter]'
 1817 |           std::__insertion_sort(__first, __first + int(_S_threshold), __comp);
      |           ~~~~~~~~~~~~~~~~~~~~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
/usr/include/c++/14/bits/stl_algo.h:1908:31:   required from 'void std::__sort(_RandomAccessIterator, _RandomAccessIterator, _Compare) [with _RandomAccessIterator = __gnu_cxx::__normal_iterator<complex<double>*, vector<complex<double> > >; _Compare = __gnu_cxx::__ops::_Iter_less_iter]'
 1908 |           std::__final_insertion_sort(__first, __last, __comp);
      |           ~~~~~~~~~~~~~~~~~~~~~~~~~~~^~~~~~~~~~~~~~~~~~~~~~~~~
/usr/include/c++/14/bits/stl_algo.h:4772:18:   required from 'void std::sort(_RAIter, _RAIter) [with _RAIter = __gnu_cxx::__normal_iterator<complex<double>*, vector<complex<double> > >]'
 4772 |       std::__sort(__first, __last, __gnu_cxx::__ops::__iter_less_iter());
      |       ~~~~~~~~~~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~