#line 2 "/Users/noya2/Desktop/Noya2_library/template/template.hpp" using namespace std; #include #line 1 "/Users/noya2/Desktop/Noya2_library/template/inout_old.hpp" namespace noya2 { template ostream &operator<<(ostream &os, const pair &p){ os << p.first << " " << p.second; return os; } template istream &operator>>(istream &is, pair &p){ is >> p.first >> p.second; return is; } template ostream &operator<<(ostream &os, const vector &v){ int s = (int)v.size(); for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i]; return os; } template istream &operator>>(istream &is, vector &v){ for (auto &x : v) is >> x; return is; } void in() {} template void in(T &t, U &...u){ cin >> t; in(u...); } void out() { cout << "\n"; } template void out(const T &t, const U &...u){ cout << t; if (sizeof...(u)) cout << sep; out(u...); } template void out(const vector> &vv){ int s = (int)vv.size(); for (int i = 0; i < s; i++) out(vv[i]); } struct IoSetup { IoSetup(){ cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); cerr << fixed << setprecision(7); } } iosetup_noya2; } // namespace noya2 #line 1 "/Users/noya2/Desktop/Noya2_library/template/const.hpp" namespace noya2{ const int iinf = 1'000'000'007; const long long linf = 2'000'000'000'000'000'000LL; const long long mod998 = 998244353; const long long mod107 = 1000000007; const long double pi = 3.14159265358979323; const vector dx = {0,1,0,-1,1,1,-1,-1}; const vector dy = {1,0,-1,0,1,-1,-1,1}; const string ALP = "ABCDEFGHIJKLMNOPQRSTUVWXYZ"; const string alp = "abcdefghijklmnopqrstuvwxyz"; const string NUM = "0123456789"; void yes(){ cout << "Yes\n"; } void no(){ cout << "No\n"; } void YES(){ cout << "YES\n"; } void NO(){ cout << "NO\n"; } void yn(bool t){ t ? yes() : no(); } void YN(bool t){ t ? YES() : NO(); } } // namespace noya2 #line 1 "/Users/noya2/Desktop/Noya2_library/template/utils.hpp" namespace noya2{ unsigned long long inner_binary_gcd(unsigned long long a, unsigned long long b){ if (a == 0 || b == 0) return a + b; int n = __builtin_ctzll(a); a >>= n; int m = __builtin_ctzll(b); b >>= m; while (a != b) { int mm = __builtin_ctzll(a - b); bool f = a > b; unsigned long long c = f ? a : b; b = f ? b : a; a = (c - b) >> mm; } return a << min(n, m); } template T gcd_fast(T a, T b){ return static_cast(inner_binary_gcd(abs(a),abs(b))); } long long sqrt_fast(long long n) { if (n <= 0) return 0; long long x = sqrt(n); while ((x + 1) * (x + 1) <= n) x++; while (x * x > n) x--; return x; } template T floor_div(const T n, const T d) { assert(d != 0); return n / d - static_cast((n ^ d) < 0 && n % d != 0); } template T ceil_div(const T n, const T d) { assert(d != 0); return n / d + static_cast((n ^ d) >= 0 && n % d != 0); } template void uniq(vector &v){ sort(v.begin(),v.end()); v.erase(unique(v.begin(),v.end()),v.end()); } template inline bool chmin(T &x, U y) { return (y < x) ? (x = y, true) : false; } template inline bool chmax(T &x, U y) { return (x < y) ? (x = y, true) : false; } template inline bool range(T l, T x, T r){ return l <= x && x < r; } } // namespace noya2 #line 8 "/Users/noya2/Desktop/Noya2_library/template/template.hpp" #define rep(i,n) for (int i = 0; i < (int)(n); i++) #define repp(i,m,n) for (int i = (m); i < (int)(n); i++) #define reb(i,n) for (int i = (int)(n-1); i >= 0; i--) #define all(v) (v).begin(),(v).end() using ll = long long; using ld = long double; using uint = unsigned int; using ull = unsigned long long; using pii = pair; using pll = pair; using pil = pair; using pli = pair; namespace noya2{ /* ~ (. _________ . /) */ } using namespace noya2; #line 2 "c.cpp" #line 2 "/Users/noya2/Desktop/Noya2_library/graph/graph_query.hpp" #line 2 "/Users/noya2/Desktop/Noya2_library/data_structure/csr.hpp" #line 4 "/Users/noya2/Desktop/Noya2_library/data_structure/csr.hpp" #include #line 7 "/Users/noya2/Desktop/Noya2_library/data_structure/csr.hpp" namespace noya2::internal { template struct csr final { csr () {} csr (int _n) : n(_n) {} csr (int _n, int m) : n(_n){ start.reserve(m); elist.reserve(m); } // ACL style constructor (do not have to call build) csr (int _n, const std::vector> &idx_elem) : n(_n), start(_n + 2), elist(idx_elem.size()) { for (auto &[i, e] : idx_elem){ start[i + 2]++; } for (int i = 1; i < n; i++){ start[i + 2] += start[i + 1]; } for (auto &[i, e] : idx_elem){ elist[start[i + 1]++] = e; } prepared = true; } int add(int idx, E elem){ int eid = start.size(); start.emplace_back(idx); elist.emplace_back(elem); return eid; } void build(){ if (prepared) return ; int m = start.size(); std::vector nelist(m); std::vector nstart(n + 2, 0); for (int i = 0; i < m; i++){ nstart[start[i] + 2]++; } for (int i = 1; i < n; i++){ nstart[i + 2] += nstart[i + 1]; } for (int i = 0; i < m; i++){ nelist[nstart[start[i] + 1]++] = elist[i]; } swap(elist,nelist); swap(start,nstart); prepared = true; } const auto operator[](int idx) const { return std::ranges::subrange(elist.begin()+start[idx],elist.begin()+start[idx+1]); } auto operator[](int idx){ return std::ranges::subrange(elist.begin()+start[idx],elist.begin()+start[idx+1]); } const auto operator()(int idx, int l, int r) const { return std::ranges::subrange(elist.begin()+start[idx]+l,elist.begin()+start[idx]+r); } auto operator()(int idx, int l, int r){ return std::ranges::subrange(elist.begin()+start[idx]+l,elist.begin()+start[idx]+r); } int n; std::vector start; std::vector elist; bool prepared = false; }; } // namespace noya2::internal #line 5 "/Users/noya2/Desktop/Noya2_library/graph/graph_query.hpp" namespace noya2 { template struct Graph { int n, m; internal::csr> g; Cost dist_inf = numeric_limits::max() / 2; Graph (int n_ = 0) : n(n_), m(-1), g(n_) {} Graph (int n_, int m_) : n(n_), m(m_), g(n_,m_) {} // 有向辺を追加 (無向辺ではないことに注意!) int add_edge(int u, int v, Cost cost = 1){ int id = g.add(u,pair(v,cost)); if (id == m-1) build(); return id; } void build(){ g.build(); } void set_inf(Cost new_inf){ dist_inf = new_inf; } vector dijkstra(int s){ vector dist(n,dist_inf); dist[s] = 0; using P = pair; priority_queue,greater

> pque; pque.push(P(0,s)); while (!pque.empty()){ auto [d, v] = pque.top(); pque.pop(); if (dist[v] < d) continue; for (auto [u, c] : g[v]){ if (chmin(dist[u],d+c)){ pque.push(P(dist[u],u)); } } } return dist; } vector reconstruct(int s, int t, const vector &dist){ if (dist[t] == dist_inf) return {}; vector from(n,-1); queue que; que.push(s); while (!que.empty()){ int v = que.front(); que.pop(); for (auto [u, c] : g[v]){ if (from[u] == -1 && dist[u] == dist[v] + c){ from[u] = v; que.push(u); } } } vector ans = {t}; while (t != s){ t = from[t]; ans.emplace_back(t); } reverse(all(ans)); return ans; } vector bfs01(int s){ vector dist(n,dist_inf); dist[s] = 0; deque que; que.push_back(s); while (!que.empty()){ int v = que.front(); que.pop_front(); for (auto [u, c] : g[v]){ if (chmin(dist[u],dist[v]+c)){ if (c == 0) que.push_front(u); else que.push_back(u); } } } return dist; } vector bfs1(int s){ vector dist(n,dist_inf); dist[s] = 0; queue que; que.push(s); while (!que.empty()){ int v = que.front(); que.pop(); for (auto [u, c] : g[v]){ if (chmin(dist[u],dist[v]+c)){ que.push(u); } } } return dist; } vector bellman_ford(int s, bool &ng_cycle){ vector dist(n,dist_inf); vector ng; dist[s] = 0; int tm = 0; while (tm < n){ bool finish = true; for (int v = 0; v < n; v++){ if (dist[v] == dist_inf) continue; for (auto [u, c] : g[v]){ if (chmin(dist[u],dist[v]+c)){ finish = false; if (tm == n-1) ng.emplace_back(u); } } } if (finish) break; tm++; } ng_cycle = (tm == n); if (ng_cycle){ for (auto v : ng) dist[v] = -dist_inf; tm = n; while (tm--){ for (int v = 0; v < n; v++){ if (dist[v] != -dist_inf) continue; for (auto [u, c] : g[v]){ dist[u] = -dist_inf; } } } } return dist; } vector> warshall_floyd(){ vector> dist(n,vector(n,dist_inf)); rep(v,n){ dist[v][v] = 0; for (auto [u, c] : g[v]){ chmin(dist[v][u],c); } } rep(k,n) rep(i,n) rep(j,n){ chmin(dist[i][j],dist[i][k]+dist[k][j]); } return dist; } const auto operator[](int idx) const { return g[idx]; } auto operator[](int idx) { return g[idx]; } }; } // namespace noya2 #line 4 "c.cpp" void solve(){ ll n; in(n); ll a, b, c; in(a,b,c); vector fact(n,1); vector big(n,false); rep(i,n-1){ fact[i+1] = fact[i] * (i+1) % n; if (big[i]){ big[i+1] = true; } else { if (fact[i] * (i+1) >= n){ big[i+1] = true; } } } Graph g(n*2); g.add_edge(0,n,0); for (ll x = 1; x < n; x++){ g.add_edge(x, x+1, a); ll y = x*x%n, cost = b*b, xp = (x*x >= n ? -1 : x*x); while (cost <= a*n){ if (xp == -1){ g.add_edge(x, n + y, cost); } else { g.add_edge(x, y, cost); xp *= x; if (xp >= n){ xp = -1; } } y = y*x%n; cost *= b; } g.add_edge(x,fact[x] + (big[x] ? n : 0),c); } for (ll x = n+1; x < 2*n; x++){ g.add_edge(x, x == 2*n-1 ? n : x+1, a); ll y = x*x%n, cost = b*b; while (cost <= a*n){ g.add_edge(x,n+y,cost); y = y*x%n; cost *= b; } g.add_edge(x,n,c); } g.build(); out(g.dijkstra(1)[n]); } int main(){ int t = 1; //in(t); while (t--) { solve(); } }