#include #ifdef _MSC_VER # include #else # include #endif #include #include namespace suisen { // ! utility template using constraints_t = std::enable_if_t, std::nullptr_t>; template constexpr decltype(auto) constexpr_if(Then&& then, OrElse&& or_else) { if constexpr (cond_v) { return std::forward(then); } else { return std::forward(or_else); } } // ! function template using is_same_as_invoke_result = std::is_same, ReturnType>; template using is_uni_op = is_same_as_invoke_result; template using is_bin_op = is_same_as_invoke_result; template using is_comparator = std::is_same, bool>; // ! integral template >> constexpr int bit_num = std::numeric_limits>::digits; template struct is_nbit { static constexpr bool value = bit_num == n; }; template static constexpr bool is_nbit_v = is_nbit::value; // ? template struct safely_multipliable {}; template <> struct safely_multipliable { using type = long long; }; template <> struct safely_multipliable { using type = __int128_t; }; template <> struct safely_multipliable { using type = unsigned long long; }; template <> struct safely_multipliable { using type = __uint128_t; }; template <> struct safely_multipliable { using type = __uint128_t; }; template <> struct safely_multipliable { using type = float; }; template <> struct safely_multipliable { using type = double; }; template <> struct safely_multipliable { using type = long double; }; template using safely_multipliable_t = typename safely_multipliable::type; template struct rec_value_type { using type = T; }; template struct rec_value_type> { using type = typename rec_value_type::type; }; template using rec_value_type_t = typename rec_value_type::type; } // namespace suisen // ! type aliases using i128 = __int128_t; using u128 = __uint128_t; template using pq_greater = std::priority_queue, std::greater>; template using umap = std::unordered_map; // ! macros (capital: internal macro) #define OVERLOAD2(_1,_2,name,...) name #define OVERLOAD3(_1,_2,_3,name,...) name #define OVERLOAD4(_1,_2,_3,_4,name,...) name #define REP4(i,l,r,s) for(std::remove_reference_t>i=(l);i<(r);i+=(s)) #define REP3(i,l,r) REP4(i,l,r,1) #define REP2(i,n) REP3(i,0,n) #define REPINF3(i,l,s) for(std::remove_reference_t>i=(l);;i+=(s)) #define REPINF2(i,l) REPINF3(i,l,1) #define REPINF1(i) REPINF2(i,0) #define RREP4(i,l,r,s) for(std::remove_reference_t>i=(l)+fld((r)-(l)-1,s)*(s);i>=(l);i-=(s)) #define RREP3(i,l,r) RREP4(i,l,r,1) #define RREP2(i,n) RREP3(i,0,n) #define rep(...) OVERLOAD4(__VA_ARGS__, REP4 , REP3 , REP2 )(__VA_ARGS__) #define rrep(...) OVERLOAD4(__VA_ARGS__, RREP4 , RREP3 , RREP2 )(__VA_ARGS__) #define repinf(...) OVERLOAD3(__VA_ARGS__, REPINF3, REPINF2, REPINF1)(__VA_ARGS__) #define CAT_I(a, b) a##b #define CAT(a, b) CAT_I(a, b) #define UNIQVAR(tag) CAT(tag, __LINE__) #define loop(n) for (std::remove_reference_t> UNIQVAR(loop_variable) = n; UNIQVAR(loop_variable) --> 0;) #define all(iterable) std::begin(iterable), std::end(iterable) #define input(type, ...) type __VA_ARGS__; read(__VA_ARGS__) #ifdef LOCAL # define debug(...) debug_internal(#__VA_ARGS__, __VA_ARGS__) template void debug_internal(const char* s, T&& first, Args&&... args) { constexpr const char* prefix = "[\033[32mDEBUG\033[m] "; constexpr const char* open_brakets = sizeof...(args) == 0 ? "" : "("; constexpr const char* close_brakets = sizeof...(args) == 0 ? "" : ")"; std::cerr << prefix << open_brakets << s << close_brakets << ": " << open_brakets << std::forward(first); ((std::cerr << ", " << std::forward(args)), ...); std::cerr << close_brakets << "\n"; } #else # define debug(...) void(0) #endif // ! I/O utilities // __int128_t std::ostream& operator<<(std::ostream& dest, __int128_t value) { std::ostream::sentry s(dest); if (s) { __uint128_t tmp = value < 0 ? -value : value; char buffer[128]; char* d = std::end(buffer); do { --d; *d = "0123456789"[tmp % 10]; tmp /= 10; } while (tmp != 0); if (value < 0) { --d; *d = '-'; } int len = std::end(buffer) - d; if (dest.rdbuf()->sputn(d, len) != len) { dest.setstate(std::ios_base::badbit); } } return dest; } // __uint128_t std::ostream& operator<<(std::ostream& dest, __uint128_t value) { std::ostream::sentry s(dest); if (s) { char buffer[128]; char* d = std::end(buffer); do { --d; *d = "0123456789"[value % 10]; value /= 10; } while (value != 0); int len = std::end(buffer) - d; if (dest.rdbuf()->sputn(d, len) != len) { dest.setstate(std::ios_base::badbit); } } return dest; } // pair template std::ostream& operator<<(std::ostream& out, const std::pair& a) { return out << a.first << ' ' << a.second; } // tuple template std::ostream& operator<<(std::ostream& out, const std::tuple& a) { if constexpr (N >= std::tuple_size_v>) { return out; } else { out << std::get(a); if constexpr (N + 1 < std::tuple_size_v>) { out << ' '; } return operator<<(out, a); } } // vector template std::ostream& operator<<(std::ostream& out, const std::vector& a) { for (auto it = a.begin(); it != a.end();) { out << *it; if (++it != a.end()) out << ' '; } return out; } // array template std::ostream& operator<<(std::ostream& out, const std::array& a) { for (auto it = a.begin(); it != a.end();) { out << *it; if (++it != a.end()) out << ' '; } return out; } inline void print() { std::cout << '\n'; } template inline void print(const Head& head, const Tail &...tails) { std::cout << head; if (sizeof...(tails)) std::cout << ' '; print(tails...); } template auto print_all(const Iterable& v, std::string sep = " ", std::string end = "\n") -> decltype(std::cout << *v.begin(), void()) { for (auto it = v.begin(); it != v.end();) { std::cout << *it; if (++it != v.end()) std::cout << sep; } std::cout << end; } __int128_t parse_i128(std::string& s) { __int128_t ret = 0; for (int i = 0; i < int(s.size()); i++) if ('0' <= s[i] and s[i] <= '9') ret = 10 * ret + s[i] - '0'; if (s[0] == '-') ret = -ret; return ret; } __uint128_t parse_u128(std::string& s) { __uint128_t ret = 0; for (int i = 0; i < int(s.size()); i++) if ('0' <= s[i] and s[i] <= '9') ret = 10 * ret + s[i] - '0'; return ret; } // __int128_t std::istream& operator>>(std::istream& in, __int128_t& v) { std::string s; in >> s; v = parse_i128(s); return in; } // __uint128_t std::istream& operator>>(std::istream& in, __uint128_t& v) { std::string s; in >> s; v = parse_u128(s); return in; } // pair template std::istream& operator>>(std::istream& in, std::pair& a) { return in >> a.first >> a.second; } // tuple template std::istream& operator>>(std::istream& in, std::tuple& a) { if constexpr (N >= std::tuple_size_v>) { return in; } else { return operator>>(in >> std::get(a), a); } } // vector template std::istream& operator>>(std::istream& in, std::vector& a) { for (auto it = a.begin(); it != a.end(); ++it) in >> *it; return in; } // array template std::istream& operator>>(std::istream& in, std::array& a) { for (auto it = a.begin(); it != a.end(); ++it) in >> *it; return in; } template void read(Args &...args) { (std::cin >> ... >> args); } // ! integral utilities // Returns pow(-1, n) template constexpr inline int pow_m1(T n) { return -(n & 1) | 1; } // Returns pow(-1, n) template <> constexpr inline int pow_m1(bool n) { return -int(n) | 1; } // Returns floor(x / y) template constexpr inline T fld(const T x, const T y) { return (x ^ y) >= 0 ? x / y : (x - (y + pow_m1(y >= 0))) / y; } template constexpr inline T cld(const T x, const T y) { return (x ^ y) <= 0 ? x / y : (x + (y + pow_m1(y >= 0))) / y; } template > = nullptr> __attribute__((target("popcnt"))) constexpr inline int popcount(const T x) { return _mm_popcnt_u32(x); } template > = nullptr> __attribute__((target("popcnt"))) constexpr inline int popcount(const T x) { return _mm_popcnt_u32(x); } template > = nullptr> __attribute__((target("popcnt"))) constexpr inline int popcount(const T x) { return _mm_popcnt_u64(x); } template > = nullptr> constexpr inline int count_lz(const T x) { return x ? __builtin_clz(x) : suisen::bit_num; } template > = nullptr> constexpr inline int count_lz(const T x) { return x ? __builtin_clz(x) : suisen::bit_num; } template > = nullptr> constexpr inline int count_lz(const T x) { return x ? __builtin_clzll(x) : suisen::bit_num; } template > = nullptr> constexpr inline int count_tz(const T x) { return x ? __builtin_ctz(x) : suisen::bit_num; } template > = nullptr> constexpr inline int count_tz(const T x) { return x ? __builtin_ctz(x) : suisen::bit_num; } template > = nullptr> constexpr inline int count_tz(const T x) { return x ? __builtin_ctzll(x) : suisen::bit_num; } template constexpr inline int floor_log2(const T x) { return suisen::bit_num -1 - count_lz(x); } template constexpr inline int ceil_log2(const T x) { return floor_log2(x) + ((x & -x) != x); } template constexpr inline int kth_bit(const T x, const unsigned int k) { return (x >> k) & 1; } template constexpr inline int parity(const T x) { return popcount(x) & 1; } // ! container template > = nullptr> auto priqueue_comp(const Comparator comparator) { return std::priority_queue, Comparator>(comparator); } template auto isize(const Iterable& iterable) -> decltype(int(iterable.size())) { return iterable.size(); } template > = nullptr> auto generate_vector(int n, Gen generator) { std::vector v(n); for (int i = 0; i < n; ++i) v[i] = generator(i); return v; } template auto generate_range_vector(T l, T r) { return generate_vector(r - l, [l](int i) { return l + i; }); } template auto generate_range_vector(T n) { return generate_range_vector(0, n); } template void sort_unique_erase(std::vector& a) { std::sort(a.begin(), a.end()); a.erase(std::unique(a.begin(), a.end()), a.end()); } template auto foreach_adjacent_values(InputIterator first, InputIterator last, BiConsumer f) -> decltype(f(*first++, *last), void()) { if (first != last) for (auto itr = first, itl = itr++; itr != last; itl = itr++) f(*itl, *itr); } template auto foreach_adjacent_values(Container c, BiConsumer f) -> decltype(c.begin(), c.end(), void()) { foreach_adjacent_values(c.begin(), c.end(), f); } // ! other utilities // x <- min(x, y). returns true iff `x` has chenged. template inline bool chmin(T& x, const T& y) { if (y >= x) return false; x = y; return true; } // x <- max(x, y). returns true iff `x` has chenged. template inline bool chmax(T& x, const T& y) { if (y <= x) return false; x = y; return true; } template , std::nullptr_t> = nullptr> std::string bin(T val, int bit_num = -1) { std::string res; if (bit_num >= 0) { for (int bit = bit_num; bit-- > 0;) res += '0' + ((val >> bit) & 1); } else { for (; val; val >>= 1) res += '0' + (val & 1); std::reverse(res.begin(), res.end()); } return res; } template , std::nullptr_t> = nullptr> std::vector digits_low_to_high(T val, T base = 10) { std::vector res; for (; val; val /= base) res.push_back(val % base); if (res.empty()) res.push_back(T{ 0 }); return res; } template , std::nullptr_t> = nullptr> std::vector digits_high_to_low(T val, T base = 10) { auto res = digits_low_to_high(val, base); std::reverse(res.begin(), res.end()); return res; } template std::string join(const std::vector& v, const std::string& sep, const std::string& end) { std::ostringstream ss; for (auto it = v.begin(); it != v.end();) { ss << *it; if (++it != v.end()) ss << sep; } ss << end; return ss.str(); } namespace suisen {} using namespace suisen; using namespace std; struct io_setup { io_setup(int precision = 20) { std::ios::sync_with_stdio(false); std::cin.tie(nullptr); std::cout << std::fixed << std::setprecision(precision); } } io_setup_{}; // ! code from here #include #include #include #include #include #include #include namespace suisen { // // Returns pow(-1, n) // template // constexpr inline int pow_m1(T n) { // return -(n & 1) | 1; // } // // Returns pow(-1, n) // template <> // constexpr inline int pow_m1(bool n) { // return -int(n) | 1; // } // // Returns floor(x / y) // template // constexpr inline T fld(const T x, const T y) { // return (x ^ y) >= 0 ? x / y : (x - (y + pow_m1(y >= 0))) / y; // } // // Returns ceil(x / y) // template // constexpr inline T cld(const T x, const T y) { // return (x ^ y) <= 0 ? x / y : (x + (y + pow_m1(y >= 0))) / y; // } /** * O(sqrt(n)) * Returns a vector of { prime, index }. * It is guaranteed that `prime` is ascending. */ template std::vector> factorize(T n) { static constexpr std::array primes{ 2, 3, 5, 7, 11, 13 }; static constexpr int next_prime = 17; static constexpr int siz = std::array{ 1, 2, 8, 48, 480, 5760, 92160 } [primes.size() - 1] ; static constexpr int period = [] { int res = 1; for (auto e : primes) res *= e; return res; }(); static constexpr struct S : public std::array { constexpr S() { for (int i = next_prime, j = 0; i < period + next_prime; i += 2) { bool ok = true; for (int p : primes) ok &= i % p > 0; if (ok) (*this)[j++] = i - next_prime; } } } s{}; assert(n > 0); std::vector> res; auto f = [&res, &n](int p) { if (n % p) return; int cnt = 0; do n /= p, ++cnt; while (n % p == 0); res.emplace_back(p, cnt); }; for (int p : primes) f(p); for (T b = next_prime; b * b <= n; b += period) { for (int offset : s) f(b + offset); } if (n != 1) res.emplace_back(n, 1); return res; } /** * O(sigma(n)) * Returns a vector that contains all divisors of `n`. * It is NOT guaranteed that the vector is sorted. */ template std::vector divisors(const std::vector>& factorized) { std::vector res{ 1 }; for (auto [p, c] : factorized) { for (int i = 0, sz = res.size(); i < sz; ++i) { T d = res[i]; for (int j = 0; j < c; ++j) res.push_back(d *= p); } } return res; } /** * O(sqrt(n)) * Returns a vector that contains all divisors of `n`. * It is NOT guaranteed that the vector is sorted. */ template > = nullptr> std::vector divisors(T n) { return divisors(factorize(n)); } template T totient(T n) { for (const auto& [p, _] : factorize(n)) n /= p, n *= p - 1; return n; } std::vector totient_table(int n) { std::vector res(n + 1); for (int i = 0; i <= n; ++i) res[i] = (i & 1) == 0 ? i >> 1 : i; for (int p = 3; p * p <= n; p += 2) { if (res[p] != p) continue; for (int q = p; q <= n; q += p) res[q] /= p, res[q] *= p - 1; } return res; } // Returns { l, r } := min_max { x>0 | fld(n,x)=q }. template > = nullptr> std::optional> same_fld_denominators_positive(T n, T q, T max_val = std::numeric_limits::max()) { T l, r; if (q >= 0) { if (n < 0) return std::nullopt; // cld(n + 1, q + 1) <= x <= fld(n, q) l = (n + 1 + q) / (q + 1), r = q == 0 ? max_val : std::min(max_val, n / q); } else { if (n >= 0) return std::nullopt; // cld(n, q) <= x <= fld(n + 1, q + 1) l = (n + q + 1) / q, r = q == -1 ? max_val : std::min(max_val, (n + 1) / (q + 1)); } if (l <= r) return std::make_pair(l, r); else return std::nullopt; } // Returns { l, r } := min_max { x<0 | fld(n,x)=q }. template > = nullptr> std::optional> same_fld_denominators_negative(T n, T q, T min_val = std::numeric_limits::min()) { T l, r; if (q >= 0) { if (n > 0) return std::nullopt; // cld(n, q) <= x <= fld(n - 1, q + 1) l = q == 0 ? min_val : std::max(min_val, n / q), r = (n - 1 - q) / (q + 1); } else { if (n <= 0) return std::nullopt; // cld(n - 1, q + 1) <= x <= fld(n, q) l = q == -1 ? min_val : std::max(min_val, (n - 1) / (q + 1)), r = (n - q - 1) / q; } if (l <= r) return std::make_pair(l, r); else return std::nullopt; } // Returns { l, r } := min_max { x>0 | cld(n,x)=q }. template > = nullptr> std::optional> same_cld_denominators_positive(T n, T q, T max_val = std::numeric_limits::max()) { T l, r; if (q > 0) { if (n <= 0) return std::nullopt; l = (n + q - 1) / q, r = q == 1 ? max_val : std::min(max_val, (n - 1) / (q - 1)); } else { if (n > 0) return std::nullopt; l = (n - 1 + q) / (q - 1), r = q == 0 ? max_val : std::min(max_val, n / q); } if (l <= r) return std::make_pair(l, r); else return std::nullopt; } // Returns { l, r } := min_max { x<0 | cld(n,x)=q }. template > = nullptr> std::optional> same_cld_denominators_negative(T n, T q, T min_val = std::numeric_limits::min()) { T l, r; if (q > 0) { if (n >= 0) return std::nullopt; l = q == 1 ? min_val : std::max(min_val, (n + 1) / (q - 1)), r = (n - q + 1) / q; } else { if (n < 0) return std::nullopt; l = q == 0 ? min_val : std::max(min_val, n / q), r = (n + 1 - q) / (q - 1); } if (l <= r) return std::make_pair(l, r); else return std::nullopt; } /** * O(sqrt(n)). * Returns vector of { l : T, r : T, q : T } s.t. let S be { d | n / d = q }, l = min S and r = max S. * It is guaranteed that `l`, `r` is ascending (i.e. `q` is descending). */ template > = nullptr> auto enumerate_quotients(T n) { assert(0 <= n); std::vector> res; for (T l = 1, r = 1; l <= n; l = r + 1) { T q = n / l; res.emplace_back(l, r = n / q, q); } return res; } /** * Template Parameter: * - vector or array * * Time Complexity: O(|vs| * Sum_{v in vs} sqrt(v)) * * Returns vector of { l : T, r : T, qs : Container } s.t. let S be { d | vs[i] / d = qs[i] (for all i) }, l = min S and r = max S * It is guaranteed that `l`, `r` is ascending (i.e. for all `i`, `q[i]` is descending). */ template std::vector> enumerate_multiple_quotients(const Container& vs) { using T = typename Container::value_type; static_assert(std::is_integral_v); int n = vs.size(); T max_val = *std::max_element(vs.begin(), vs.end()); assert(*std::min_element(vs.begin(), vs.end()) >= 0); std::vector> res; for (T l = 1, r = 1; l <= max_val; l = r + 1) { Container qs{}; if constexpr (std::is_same_v>) qs.resize(n); r = std::numeric_limits::max(); for (int i = 0; i < n; ++i) { qs[i] = vs[i] / l; r = std::min(r, qs[i] == 0 ? std::numeric_limits::max() : vs[i] / qs[i]); } res.emplace_back(l, r, std::move(qs)); } return res; } template , std::nullptr_t> = nullptr> T floor_kth_root(T x, int k) { if (k == 1 or x == 0 or x == 1) return x; if (k >= 64) return 1; if (k == 2) return sqrtl(x); // if (k == 3) return cbrtl(x); T res = powl(x, nextafterl(1 / (long double) k, 0)); while (powl(res + 1, k) <= x) ++res; return res; } } // namespace suisen #include namespace suisen { template class dijkstra { public: template dijkstra(unsigned int n, Transition transition, unsigned int src) : _src(src) { _par.resize(n); _dist.assign(n, UNREACHABLE); _dist[src] = 0; using state = std::pair; std::priority_queue, std::greater> pq; pq.emplace(0, src); auto g = [&](unsigned int u) { return [&, u](unsigned int v, Cost new_cost) { if (new_cost < _dist[v]) pq.emplace(_dist[v] = new_cost, v), _par[v] = u; }; }; while (pq.size()) { auto [du, u] = pq.top(); pq.pop(); if (du == _dist[u]) transition(u, du, g(u)); } } dijkstra(const std::vector>> &g, unsigned int src) : dijkstra(g.size(), [&](int u, Cost du, auto f) { for (auto [v, c] : g[u]) f(v, du + c); }, src) {} std::vector path_to(unsigned int t) const { assert(is_reachale(t)); std::vector path = {t}; while (t != _src) path.push_back(t = _par[t]); std::reverse(path.begin(), path.end()); return path; } Cost operator[](unsigned int t) const { return _dist[t]; } bool is_reachale (unsigned int t) const { return _dist[t] != UNREACHABLE; } bool is_unreachable(unsigned int t) const { return _dist[t] == UNREACHABLE; } private: const Cost UNREACHABLE = std::numeric_limits::max(); const unsigned int _src; std::vector _dist; std::vector _par; }; } // namespace suisen int main() { input(int, n, m); input(long long, a, b); vector c(m); read(c); auto div = divisors(n); sort(all(div)); const int k = div.size(); vector>> g(k); rep(i, k) { int ng = numeric_limits::max(); for (int v : c) if (v % div[i] == 0) { chmin(ng, v); } rep(j, i + 1, k) { if (div[j] >= ng) break; if (div[j] % div[i]) continue; long long t = div[j] / div[i]; g[i].emplace_back(j, (t - 1) * a + b); } } dijkstra dij(g, 0); if (dij.is_unreachable(k - 1)) { print(-1); } else { print(dij[k - 1] - b); } return 0; }