#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 2 "/Users/noya2/Desktop/Noya2_library/math/prime.hpp" #line 4 "/Users/noya2/Desktop/Noya2_library/math/prime.hpp" namespace noya2 { constexpr ll safe_mod(ll x, ll m) { x %= m; if (x < 0) x += m; return x; } constexpr ll pow_mod_constexpr(ll x, ll n, int m) { if (m == 1) return 0; uint _m = (uint)(m); ull r = 1; ull y = safe_mod(x, m); while (n) { if (n & 1) r = (r * y) % _m; y = (y * y) % _m; n >>= 1; } return r; } constexpr bool is_prime_constexpr(int n) { if (n <= 1) return false; if (n == 2 || n == 7 || n == 61) return true; if (n % 2 == 0) return false; ll d = n - 1; while (d % 2 == 0) d /= 2; constexpr ll bases[3] = {2, 7, 61}; for (ll a : bases) { ll t = d; ll y = pow_mod_constexpr(a, t, n); while (t != n - 1 && y != 1 && y != n - 1) { y = y * y % n; t <<= 1; } if (y != n - 1 && t % 2 == 0) { return false; } } return true; } template constexpr bool is_prime_flag = is_prime_constexpr(n); constexpr std::pair inv_gcd(long long a, long long b) { a = safe_mod(a, b); if (a == 0) return {b, 0}; long long s = b, t = a; long long m0 = 0, m1 = 1; while (t) { long long u = s / t; s -= t * u; m0 -= m1 * u; auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } if (m0 < 0) m0 += b / s; return {s, m0}; } constexpr int primitive_root_constexpr(int m) { if (m == 2) return 1; if (m == 167772161) return 3; if (m == 469762049) return 3; if (m == 754974721) return 11; if (m == 998244353) return 3; int divs[20] = {}; divs[0] = 2; int cnt = 1; int x = (m - 1) / 2; while (x % 2 == 0) x /= 2; for (int i = 3; (ll)(i)*i <= x; i += 2) { if (x % i == 0) { divs[cnt++] = i; while (x % i == 0) { x /= i; } } } if (x > 1) { divs[cnt++] = x; } for (int g = 2;; g++) { bool ok = true; for (int i = 0; i < cnt; i++) { if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) { ok = false; break; } } if (ok) return g; } } template constexpr int primitive_root_flag = primitive_root_constexpr(m); } // namespace noya2 #line 5 "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; } } } vector isprime(60); rep(i,60) if (is_prime_constexpr(i)) isprime[i] = 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); int p = 2; while (cost <= a*n){ if (isprime[p]){ if (xp == -1){ g.add_edge(x, n + y, cost); } else { g.add_edge(x, y, cost); xp *= x; if (xp >= n){ xp = -1; } } } else { if (xp != -1){ 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; int p = 2; while (cost <= a*n){ if (isprime[p]){ 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(); } }