結果
| 問題 | No.2348 Power!! (Easy) |
| ユーザー |
 suisen
|
| 提出日時 | 2023-06-09 22:22:44 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
TLE
|
| 実行時間 | - |
| コード長 | 55,212 bytes |
| コンパイル時間 | 5,478 ms |
| コンパイル使用メモリ | 369,236 KB |
| 最終ジャッジ日時 | 2025-02-14 00:00:31 |
|
ジャッジサーバーID （参考情報） |
judge2 / judge3 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 1 |
| other | AC * 4 TLE * 8 |
ソースコード
#include <bits/stdc++.h>
#ifdef _MSC_VER
# include <intrin.h>
#else
# include <x86intrin.h>
#endif
#include <limits>
#include <type_traits>
namespace suisen {
// ! utility
template <typename ...Types>
using constraints_t = std::enable_if_t<std::conjunction_v<Types...>, std::nullptr_t>;
template <bool cond_v, typename Then, typename OrElse>
constexpr decltype(auto) constexpr_if(Then&& then, OrElse&& or_else) {
if constexpr (cond_v) {
return std::forward<Then>(then);
} else {
return std::forward<OrElse>(or_else);
}
}
// ! function
template <typename ReturnType, typename Callable, typename ...Args>
using is_same_as_invoke_result = std::is_same<std::invoke_result_t<Callable, Args...>, ReturnType>;
template <typename F, typename T>
using is_uni_op = is_same_as_invoke_result<T, F, T>;
template <typename F, typename T>
using is_bin_op = is_same_as_invoke_result<T, F, T, T>;
template <typename Comparator, typename T>
using is_comparator = std::is_same<std::invoke_result_t<Comparator, T, T>, bool>;
// ! integral
template <typename T, typename = constraints_t<std::is_integral<T>>>
constexpr int bit_num = std::numeric_limits<std::make_unsigned_t<T>>::digits;
template <typename T, unsigned int n>
struct is_nbit { static constexpr bool value = bit_num<T> == n; };
template <typename T, unsigned int n>
static constexpr bool is_nbit_v = is_nbit<T, n>::value;
// ?
template <typename T>
struct safely_multipliable {};
template <>
struct safely_multipliable<int> { using type = long long; };
template <>
struct safely_multipliable<long long> { using type = __int128_t; };
template <>
struct safely_multipliable<unsigned int> { using type = unsigned long long; };
template <>
struct safely_multipliable<unsigned long int> { using type = __uint128_t; };
template <>
struct safely_multipliable<unsigned long long> { using type = __uint128_t; };
template <>
struct safely_multipliable<float> { using type = float; };
template <>
struct safely_multipliable<double> { using type = double; };
template <>
struct safely_multipliable<long double> { using type = long double; };
template <typename T>
using safely_multipliable_t = typename safely_multipliable<T>::type;
template <typename T, typename = void>
struct rec_value_type {
using type = T;
};
template <typename T>
struct rec_value_type<T, std::void_t<typename T::value_type>> {
using type = typename rec_value_type<typename T::value_type>::type;
};
template <typename T>
using rec_value_type_t = typename rec_value_type<T>::type;
} // namespace suisen
// ! type aliases
using i128 = __int128_t;
using u128 = __uint128_t;
template <typename T>
using pq_greater = std::priority_queue<T, std::vector<T>, std::greater<T>>;
// ! macros (internal)
#define DETAIL_OVERLOAD2(_1,_2,name,...) name
#define DETAIL_OVERLOAD3(_1,_2,_3,name,...) name
#define DETAIL_OVERLOAD4(_1,_2,_3,_4,name,...) name
#define DETAIL_REP4(i,l,r,s) for(std::remove_reference_t<std::remove_const_t<decltype(r)>>i=(l);i<(r);i+=(s))
#define DETAIL_REP3(i,l,r) DETAIL_REP4(i,l,r,1)
#define DETAIL_REP2(i,n) DETAIL_REP3(i,0,n)
#define DETAIL_REPINF3(i,l,s) for(std::remove_reference_t<std::remove_const_t<decltype(l)>>i=(l);;i+=(s))
#define DETAIL_REPINF2(i,l) DETAIL_REPINF3(i,l,1)
#define DETAIL_REPINF1(i) DETAIL_REPINF2(i,0)
#define DETAIL_RREP4(i,l,r,s) for(std::remove_reference_t<std::remove_const_t<decltype(r)>>i=(l)+fld((r)-(l)-1,s)*(s);i>=(l);i-=(s))
#define DETAIL_RREP3(i,l,r) DETAIL_RREP4(i,l,r,1)
#define DETAIL_RREP2(i,n) DETAIL_RREP3(i,0,n)
#define DETAIL_CAT_I(a, b) a##b
#define DETAIL_CAT(a, b) DETAIL_CAT_I(a, b)
#define DETAIL_UNIQVAR(tag) DETAIL_CAT(tag, __LINE__)
// ! macros
#define REP(...) DETAIL_OVERLOAD4(__VA_ARGS__, DETAIL_REP4 , DETAIL_REP3 , DETAIL_REP2 )(__VA_ARGS__)
#define RREP(...) DETAIL_OVERLOAD4(__VA_ARGS__, DETAIL_RREP4 , DETAIL_RREP3 , DETAIL_RREP2 )(__VA_ARGS__)
#define REPINF(...) DETAIL_OVERLOAD3(__VA_ARGS__, DETAIL_REPINF3, DETAIL_REPINF2, DETAIL_REPINF1)(__VA_ARGS__)
#define LOOP(n) for (std::remove_reference_t<std::remove_const_t<decltype(n)>> DETAIL_UNIQVAR(loop_variable) = n; DETAIL_UNIQVAR(loop_variable) --> 0;)
#define ALL(iterable) std::begin(iterable), std::end(iterable)
#define INPUT(type, ...) type __VA_ARGS__; read(__VA_ARGS__)
// ! debug
#ifdef LOCAL
# define debug(...) debug_internal(#__VA_ARGS__, __VA_ARGS__)
template <class T, class... Args>
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<T>(first);
((std::cerr << ", " << std::forward<Args>(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 <typename T, typename U>
std::ostream& operator<<(std::ostream& out, const std::pair<T, U>& a) {
return out << a.first << ' ' << a.second;
}
// tuple
template <unsigned int N = 0, typename ...Args>
std::ostream& operator<<(std::ostream& out, const std::tuple<Args...>& a) {
if constexpr (N >= std::tuple_size_v<std::tuple<Args...>>) return out;
else {
out << std::get<N>(a);
if constexpr (N + 1 < std::tuple_size_v<std::tuple<Args...>>) out << ' ';
return operator<<<N + 1>(out, a);
}
}
// vector
template <typename T>
std::ostream& operator<<(std::ostream& out, const std::vector<T>& a) {
for (auto it = a.begin(); it != a.end();) {
out << *it;
if (++it != a.end()) out << ' ';
}
return out;
}
// array
template <typename T, size_t N>
std::ostream& operator<<(std::ostream& out, const std::array<T, N>& 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 <typename Head, typename... Tail>
inline void print(const Head& head, const Tail &...tails) {
std::cout << head;
if (sizeof...(tails)) std::cout << ' ';
print(tails...);
}
template <typename Iterable>
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 stoi128(const 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 stou128(const 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 = stoi128(s);
return in;
}
// __uint128_t
std::istream& operator>>(std::istream& in, __uint128_t& v) {
std::string s;
in >> s;
v = stou128(s);
return in;
}
// pair
template <typename T, typename U>
std::istream& operator>>(std::istream& in, std::pair<T, U>& a) {
return in >> a.first >> a.second;
}
// tuple
template <unsigned int N = 0, typename ...Args>
std::istream& operator>>(std::istream& in, std::tuple<Args...>& a) {
if constexpr (N >= std::tuple_size_v<std::tuple<Args...>>) return in;
else return operator>><N + 1>(in >> std::get<N>(a), a);
}
// vector
template <typename T>
std::istream& operator>>(std::istream& in, std::vector<T>& a) {
for (auto it = a.begin(); it != a.end(); ++it) in >> *it;
return in;
}
// array
template <typename T, size_t N>
std::istream& operator>>(std::istream& in, std::array<T, N>& a) {
for (auto it = a.begin(); it != a.end(); ++it) in >> *it;
return in;
}
template <typename ...Args>
void read(Args &...args) {
(std::cin >> ... >> args);
}
// ! integral utilities
// Returns pow(-1, n)
template <typename T> constexpr inline int pow_m1(T n) {
return -(n & 1) | 1;
}
// Returns pow(-1, n)
template <> constexpr inline int pow_m1<bool>(bool n) {
return -int(n) | 1;
}
// Returns floor(x / y)
template <typename T> constexpr inline T fld(const T x, const T y) {
return (x ^ y) >= 0 ? x / y : (x - (y + pow_m1(y >= 0))) / y;
}
template <typename T> constexpr inline T cld(const T x, const T y) {
return (x ^ y) <= 0 ? x / y : (x + (y + pow_m1(y >= 0))) / y;
}
template <typename T, std::enable_if_t<std::negation_v<suisen::is_nbit<T, 64>>, std::nullptr_t> = nullptr>
__attribute__((target("popcnt"))) constexpr inline int popcount(const T x) { return _mm_popcnt_u32(x); }
template <typename T, std::enable_if_t<suisen::is_nbit_v<T, 64>, std::nullptr_t> = nullptr>
__attribute__((target("popcnt"))) constexpr inline int popcount(const T x) { return _mm_popcnt_u64(x); }
template <typename T, std::enable_if_t<std::negation_v<suisen::is_nbit<T, 64>>, std::nullptr_t> = nullptr>
constexpr inline int count_lz(const T x) { return x ? __builtin_clz(x) : suisen::bit_num<T>; }
template <typename T, std::enable_if_t<suisen::is_nbit_v<T, 64>, std::nullptr_t> = nullptr>
constexpr inline int count_lz(const T x) { return x ? __builtin_clzll(x) : suisen::bit_num<T>; }
template <typename T, std::enable_if_t<std::negation_v<suisen::is_nbit<T, 64>>, std::nullptr_t> = nullptr>
constexpr inline int count_tz(const T x) { return x ? __builtin_ctz(x) : suisen::bit_num<T>; }
template <typename T, std::enable_if_t<suisen::is_nbit_v<T, 64>, std::nullptr_t> = nullptr>
constexpr inline int count_tz(const T x) { return x ? __builtin_ctzll(x) : suisen::bit_num<T>; }
template <typename T> constexpr inline int floor_log2(const T x) { return suisen::bit_num<T> - 1 - count_lz(x); }
template <typename T> constexpr inline int ceil_log2(const T x) { return floor_log2(x) + ((x & -x) != x); }
template <typename T> constexpr inline int kth_bit(const T x, const unsigned int k) { return (x >> k) & 1; }
template <typename T> constexpr inline int parity(const T x) { return popcount(x) & 1; }
// ! container
template <typename T, typename Comparator>
auto priqueue_comp(const Comparator comparator) {
return std::priority_queue<T, std::vector<T>, Comparator>(comparator);
}
template <typename Container>
void sort_unique_erase(Container& a) {
std::sort(a.begin(), a.end());
a.erase(std::unique(a.begin(), a.end()), a.end());
}
template <typename InputIterator, typename BiConsumer>
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 <typename Container, typename BiConsumer>
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 <typename T>
inline bool chmin(T& x, const T& y) {
return y >= x ? false : (x = y, true);
}
// x <- max(x, y). returns true iff `x` has chenged.
template <typename T>
inline bool chmax(T& x, const T& y) {
return y <= x ? false : (x = y, true);
}
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::string bin(T val, int bit_num = -1) {
std::string res;
if (bit_num != -1) {
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 <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<T> digits_low_to_high(T val, T base = 10) {
std::vector<T> res;
for (; val; val /= base) res.push_back(val % base);
if (res.empty()) res.push_back(T{ 0 });
return res;
}
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
std::vector<T> 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 <typename T>
std::string join(const std::vector<T>& 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();
}
template <typename Func, typename Seq>
auto transform_to_vector(const Func &f, const Seq &s) {
std::vector<std::invoke_result_t<Func, typename Seq::value_type>> v;
v.reserve(std::size(s)), std::transform(std::begin(s), std::end(s), std::back_inserter(v), f);
return v;
}
template <typename T, typename Seq>
auto copy_to_vector(const Seq &s) {
std::vector<T> v;
v.reserve(std::size(s)), std::copy(std::begin(s), std::end(s), std::back_inserter(v));
return v;
}
template <typename Seq>
Seq concat(Seq s, const Seq &t) {
s.reserve(std::size(s) + std::size(t));
std::copy(std::begin(t), std::end(t), std::back_inserter(s));
return s;
}
template <typename Seq>
std::vector<Seq> split(const Seq s, typename Seq::value_type delim) {
std::vector<Seq> res;
for (auto itl = std::begin(s), itr = itl;; itl = ++itr) {
while (itr != std::end(s) and *itr != delim) ++itr;
res.emplace_back(itl, itr);
if (itr == std::end(s)) return res;
}
}
int digit_to_int(char c) { return c - '0'; }
int lowercase_to_int(char c) { return c - 'a'; }
int uppercase_to_int(char c) { return c - 'A'; }
std::vector<int> digit_str_to_ints(const std::string &s) {
return transform_to_vector(digit_to_int, s);
}
std::vector<int> lowercase_str_to_ints(const std::string &s) {
return transform_to_vector(lowercase_to_int, s);
}
std::vector<int> uppercase_str_to_ints(const std::string &s) {
return transform_to_vector(uppercase_to_int, s);
}
template <typename T, typename ToKey, typename CompareValue = std::less<>,
std::enable_if_t<
std::conjunction_v<
std::is_invocable<ToKey, T>,
std::is_invocable_r<bool, CompareValue, std::invoke_result_t<ToKey, T>, std::invoke_result_t<ToKey, T>
>
>, std::nullptr_t> = nullptr
>
auto comparator(const ToKey &to_key, const CompareValue &compare_value = std::less<>()) {
return [to_key, compare_value](const T& x, const T& y) { return compare_value(to_key(x), to_key(y)); };
}
template <typename ToKey, std::enable_if_t<std::is_invocable_v<ToKey, int>, std::nullptr_t> = nullptr>
std::vector<int> sorted_indices(int n, const ToKey &to_key) {
std::vector<int> p(n);
std::iota(p.begin(), p.end(), 0);
std::sort(p.begin(), p.end(), comparator<int>(to_key));
return p;
}
template <typename Compare, std::enable_if_t<std::is_invocable_r_v<bool, Compare, int, int>, std::nullptr_t> = nullptr>
std::vector<int> sorted_indices(int n, const Compare &compare) {
std::vector<int> p(n);
std::iota(p.begin(), p.end(), 0);
std::sort(p.begin(), p.end(), compare);
return p;
}
const std::string Yes = "Yes", No = "No", YES = "YES", NO = "NO";
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 <atcoder/modint>
using mint = atcoder::modint998244353;
namespace atcoder {
std::istream& operator>>(std::istream& in, mint& a) {
long long e; in >> e; a = e;
return in;
}
std::ostream& operator<<(std::ostream& out, const mint& a) {
out << a.val();
return out;
}
} // namespace atcoder
#include <cmath>
namespace suisen {
template <typename T, std::enable_if_t<std::is_integral_v<T>, 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 == 2) return ::sqrtl(x);
if (k >= 64) return 1;
T res = ::powl(x, ::nextafterl(1 / (long double) k, 0));
while (::powl(res + 1, k) <= x) ++res;
return res;
}
template <typename T, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
T ceil_kth_root(T x, int k) {
T res = floor_kth_root(x, k);
res += ::powl(res, k) < x;
return res;
}
} // namespace suisen
#include <vector>
namespace suisen {
template <typename FPSType, typename T>
std::vector<typename FPSType::value_type> multi_point_eval(const FPSType& f, const std::vector<T>& xs) {
int n = xs.size();
if (n == 0) return {};
std::vector<FPSType> seg(2 * n);
for (int i = 0; i < n; ++i) seg[n + i] = FPSType{ -xs[i], 1 };
for (int i = n - 1; i > 0; --i) seg[i] = seg[i * 2] * seg[i * 2 + 1];
seg[1] = f % seg[1];
for (int i = 2; i < 2 * n; ++i) seg[i] = seg[i / 2] % seg[i];
std::vector<typename FPSType::value_type> ys(n);
for (int i = 0; i < n; ++i) ys[i] = seg[n + i].size() ? seg[n + i][0] : 0;
return ys;
}
} // namespace suisen
#include <optional>
#include <queue>
#include <atcoder/modint>
#include <atcoder/convolution>
#include <cassert>
/**
* refernce: https://37zigen.com/tonelli-shanks-algorithm/
* calculates x s.t. x^2 = a mod p in O((log p)^2).
*/
template <typename mint>
std::optional<mint> safe_sqrt(mint a) {
static int p = mint::mod();
if (a == 0) return std::make_optional(0);
if (p == 2) return std::make_optional(a);
if (a.pow((p - 1) / 2) != 1) return std::nullopt;
mint b = 1;
while (b.pow((p - 1) / 2) == 1) ++b;
static int tlz = __builtin_ctz(p - 1), q = (p - 1) >> tlz;
mint x = a.pow((q + 1) / 2);
b = b.pow(q);
for (int shift = 2; x * x != a; ++shift) {
mint e = a.inv() * x * x;
if (e.pow(1 << (tlz - shift)) != 1) x *= b;
b *= b;
}
return std::make_optional(x);
}
/**
* calculates x s.t. x^2 = a mod p in O((log p)^2).
* if not exists, raises runtime error.
*/
template <typename mint>
auto sqrt(mint a) -> decltype(mint::mod(), mint()) {
return *safe_sqrt(a);
}
template <typename mint>
auto log(mint a) -> decltype(mint::mod(), mint()) {
assert(a == 1);
return 0;
}
template <typename mint>
auto exp(mint a) -> decltype(mint::mod(), mint()) {
assert(a == 0);
return 1;
}
template <typename mint, typename T>
auto pow(mint a, T b) -> decltype(mint::mod(), mint()) {
return a.pow(b);
}
template <typename mint>
auto inv(mint a) -> decltype(mint::mod(), mint()) {
return a.inv();
}
namespace suisen {
template <typename mint>
class inv_mods {
public:
inv_mods() {}
inv_mods(int n) { ensure(n); }
const mint& operator[](int i) const {
ensure(i);
return invs[i];
}
static void ensure(int n) {
int sz = invs.size();
if (sz < 2) invs = { 0, 1 }, sz = 2;
if (sz < n + 1) {
invs.resize(n + 1);
for (int i = sz; i <= n; ++i) invs[i] = mint(mod - mod / i) * invs[mod % i];
}
}
private:
static std::vector<mint> invs;
static constexpr int mod = mint::mod();
};
template <typename mint>
std::vector<mint> inv_mods<mint>::invs{};
template <typename mint>
std::vector<mint> get_invs(const std::vector<mint>& vs) {
const int n = vs.size();
mint p = 1;
for (auto& e : vs) {
p *= e;
assert(e != 0);
}
mint ip = p.inv();
std::vector<mint> rp(n + 1);
rp[n] = 1;
for (int i = n - 1; i >= 0; --i) {
rp[i] = rp[i + 1] * vs[i];
}
std::vector<mint> res(n);
for (int i = 0; i < n; ++i) {
res[i] = ip * rp[i + 1];
ip *= vs[i];
}
return res;
}
}
namespace suisen {
template <typename T>
struct FPSNaive : std::vector<T> {
static inline int MAX_SIZE = std::numeric_limits<int>::max() / 2;
using value_type = T;
using element_type = rec_value_type_t<T>;
using std::vector<value_type>::vector;
FPSNaive(const std::initializer_list<value_type> l) : std::vector<value_type>::vector(l) {}
FPSNaive(const std::vector<value_type>& v) : std::vector<value_type>::vector(v) {}
static void set_max_size(int n) {
FPSNaive<T>::MAX_SIZE = n;
}
const value_type operator[](int n) const {
return n <= deg() ? unsafe_get(n) : value_type{ 0 };
}
value_type& operator[](int n) {
return ensure_deg(n), unsafe_get(n);
}
int size() const {
return std::vector<value_type>::size();
}
int deg() const {
return size() - 1;
}
int normalize() {
while (size() and this->back() == value_type{ 0 }) this->pop_back();
return deg();
}
FPSNaive& cut_inplace(int n) {
if (size() > n) this->resize(std::max(0, n));
return *this;
}
FPSNaive cut(int n) const {
FPSNaive f = FPSNaive(*this).cut_inplace(n);
return f;
}
FPSNaive operator+() const {
return FPSNaive(*this);
}
FPSNaive operator-() const {
FPSNaive f(*this);
for (auto& e : f) e = -e;
return f;
}
FPSNaive& operator++() { return ++(*this)[0], * this; }
FPSNaive& operator--() { return --(*this)[0], * this; }
FPSNaive& operator+=(const value_type x) { return (*this)[0] += x, *this; }
FPSNaive& operator-=(const value_type x) { return (*this)[0] -= x, *this; }
FPSNaive& operator+=(const FPSNaive& g) {
ensure_deg(g.deg());
for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) += g.unsafe_get(i);
return *this;
}
FPSNaive& operator-=(const FPSNaive& g) {
ensure_deg(g.deg());
for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) -= g.unsafe_get(i);
return *this;
}
FPSNaive& operator*=(const FPSNaive& g) { return *this = *this * g; }
FPSNaive& operator*=(const value_type x) {
for (auto& e : *this) e *= x;
return *this;
}
FPSNaive& operator/=(const FPSNaive& g) { return *this = *this / g; }
FPSNaive& operator%=(const FPSNaive& g) { return *this = *this % g; }
FPSNaive& operator<<=(const int shamt) {
this->insert(this->begin(), shamt, value_type{ 0 });
return *this;
}
FPSNaive& operator>>=(const int shamt) {
if (shamt > size()) this->clear();
else this->erase(this->begin(), this->begin() + shamt);
return *this;
}
friend FPSNaive operator+(FPSNaive f, const FPSNaive& g) { f += g; return f; }
friend FPSNaive operator+(FPSNaive f, const value_type& x) { f += x; return f; }
friend FPSNaive operator-(FPSNaive f, const FPSNaive& g) { f -= g; return f; }
friend FPSNaive operator-(FPSNaive f, const value_type& x) { f -= x; return f; }
friend FPSNaive operator*(const FPSNaive& f, const FPSNaive& g) {
if (f.empty() or g.empty()) return FPSNaive{};
const int n = f.size(), m = g.size();
FPSNaive h(std::min(MAX_SIZE, n + m - 1));
for (int i = 0; i < n; ++i) for (int j = 0; j < m; ++j) {
if (i + j >= MAX_SIZE) break;
h.unsafe_get(i + j) += f.unsafe_get(i) * g.unsafe_get(j);
}
return h;
}
friend FPSNaive operator*(FPSNaive f, const value_type& x) { f *= x; return f; }
friend FPSNaive operator/(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).first); }
friend FPSNaive operator%(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).second); }
friend FPSNaive operator*(const value_type x, FPSNaive f) { f *= x; return f; }
friend FPSNaive operator<<(FPSNaive f, const int shamt) { f <<= shamt; return f; }
friend FPSNaive operator>>(FPSNaive f, const int shamt) { f >>= shamt; return f; }
std::pair<FPSNaive, FPSNaive> div_mod(FPSNaive g) const {
FPSNaive f = *this;
const int fd = f.normalize(), gd = g.normalize();
assert(gd >= 0);
if (fd < gd) return { FPSNaive{}, f };
if (gd == 0) return { f *= g.unsafe_get(0).inv(), FPSNaive{} };
const int k = f.deg() - gd;
value_type head_inv = g.unsafe_get(gd).inv();
FPSNaive q(k + 1);
for (int i = k; i >= 0; --i) {
value_type div = f.unsafe_get(i + gd) * head_inv;
q.unsafe_get(i) = div;
for (int j = 0; j <= gd; ++j) f.unsafe_get(i + j) -= div * g.unsafe_get(j);
}
f.cut_inplace(gd);
f.normalize();
return { q, f };
}
friend bool operator==(const FPSNaive& f, const FPSNaive& g) {
const int n = f.size(), m = g.size();
if (n < m) return g == f;
for (int i = 0; i < m; ++i) if (f.unsafe_get(i) != g.unsafe_get(i)) return false;
for (int i = m; i < n; ++i) if (f.unsafe_get(i) != 0) return false;
return true;
}
friend bool operator!=(const FPSNaive& f, const FPSNaive& g) {
return not (f == g);
}
FPSNaive mul(const FPSNaive& g, int n = -1) const {
if (n < 0) n = size();
if (this->empty() or g.empty()) return FPSNaive{};
const int m = size(), k = g.size();
FPSNaive h(std::min(n, m + k - 1));
for (int i = 0; i < m; ++i) {
for (int j = 0, jr = std::min(k, n - i); j < jr; ++j) {
h.unsafe_get(i + j) += unsafe_get(i) * g.unsafe_get(j);
}
}
return h;
}
FPSNaive diff() const {
if (this->empty()) return {};
FPSNaive g(size() - 1);
for (int i = 1; i <= deg(); ++i) g.unsafe_get(i - 1) = unsafe_get(i) * i;
return g;
}
FPSNaive intg() const {
const int n = size();
FPSNaive g(n + 1);
for (int i = 0; i < n; ++i) g.unsafe_get(i + 1) = unsafe_get(i) * invs[i + 1];
if (g.deg() > MAX_SIZE) g.cut_inplace(MAX_SIZE);
return g;
}
FPSNaive inv(int n = -1) const {
if (n < 0) n = size();
FPSNaive g(n);
const value_type inv_f0 = ::inv(unsafe_get(0));
g.unsafe_get(0) = inv_f0;
for (int i = 1; i < n; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) -= g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= inv_f0;
}
return g;
}
FPSNaive exp(int n = -1) const {
if (n < 0) n = size();
assert(unsafe_get(0) == value_type{ 0 });
FPSNaive g(n);
g.unsafe_get(0) = value_type{ 1 };
for (int i = 1; i < n; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) += j * g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= invs[i];
}
return g;
}
FPSNaive log(int n = -1) const {
if (n < 0) n = size();
assert(unsafe_get(0) == value_type{ 1 });
FPSNaive g(n);
g.unsafe_get(0) = value_type{ 0 };
for (int i = 1; i < n; ++i) {
g.unsafe_get(i) = i * (*this)[i];
for (int j = 1; j < i; ++j) g.unsafe_get(i) -= (i - j) * g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= invs[i];
}
return g;
}
FPSNaive pow(const long long k, int n = -1) const {
if (n < 0) n = size();
if (k == 0) {
FPSNaive res(n);
res[0] = 1;
return res;
}
int z = 0;
while (z < size() and unsafe_get(z) == value_type{ 0 }) ++z;
if (z == size() or z > (n - 1) / k) return FPSNaive(n, 0);
const int m = n - z * k;
FPSNaive g(m);
const value_type inv_f0 = ::inv(unsafe_get(z));
g.unsafe_get(0) = unsafe_get(z).pow(k);
for (int i = 1; i < m; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) += (element_type{ k } *j - (i - j)) * g.unsafe_get(i - j) * (*this)[z + j];
g.unsafe_get(i) *= inv_f0 * invs[i];
}
g <<= z * k;
return g;
}
std::optional<FPSNaive> safe_sqrt(int n = -1) const {
if (n < 0) n = size();
int dl = 0;
while (dl < size() and unsafe_get(dl) == value_type{ 0 }) ++dl;
if (dl == size()) return FPSNaive(n, 0);
if (dl & 1) return std::nullopt;
const int m = n - dl / 2;
FPSNaive g(m);
auto opt_g0 = ::safe_sqrt((*this)[dl]);
if (not opt_g0.has_value()) return std::nullopt;
g.unsafe_get(0) = *opt_g0;
value_type inv_2g0 = ::inv(2 * g.unsafe_get(0));
for (int i = 1; i < m; ++i) {
g.unsafe_get(i) = (*this)[dl + i];
for (int j = 1; j < i; ++j) g.unsafe_get(i) -= g.unsafe_get(j) * g.unsafe_get(i - j);
g.unsafe_get(i) *= inv_2g0;
}
g <<= dl / 2;
return g;
}
FPSNaive sqrt(int n = -1) const {
if (n < 0) n = size();
return *safe_sqrt(n);
}
value_type eval(value_type x) const {
value_type y = 0;
for (int i = size() - 1; i >= 0; --i) y = y * x + unsafe_get(i);
return y;
}
private:
static inline inv_mods<element_type> invs;
void ensure_deg(int d) {
if (deg() < d) this->resize(d + 1, value_type{ 0 });
}
const value_type& unsafe_get(int i) const {
return std::vector<value_type>::operator[](i);
}
value_type& unsafe_get(int i) {
return std::vector<value_type>::operator[](i);
}
};
} // namespace suisen
template <typename mint>
suisen::FPSNaive<mint> sqrt(suisen::FPSNaive<mint> a) {
return a.sqrt();
}
template <typename mint>
suisen::FPSNaive<mint> log(suisen::FPSNaive<mint> a) {
return a.log();
}
template <typename mint>
suisen::FPSNaive<mint> exp(suisen::FPSNaive<mint> a) {
return a.exp();
}
template <typename mint, typename T>
suisen::FPSNaive<mint> pow(suisen::FPSNaive<mint> a, T b) {
return a.pow(b);
}
template <typename mint>
suisen::FPSNaive<mint> inv(suisen::FPSNaive<mint> a) {
return a.inv();
}
namespace suisen {
template <typename mint, atcoder::internal::is_static_modint_t<mint>* = nullptr>
struct FormalPowerSeries : std::vector<mint> {
using base_type = std::vector<mint>;
using value_type = typename base_type::value_type;
using base_type::vector;
FormalPowerSeries(const std::initializer_list<value_type> l) : std::vector<value_type>::vector(l) {}
FormalPowerSeries(const std::vector<value_type>& v) : std::vector<value_type>::vector(v) {}
int size() const noexcept {
return base_type::size();
}
int deg() const noexcept {
return size() - 1;
}
void ensure(int n) {
if (size() < n) this->resize(n);
}
value_type safe_get(int d) const {
return d <= deg() ? (*this)[d] : 0;
}
value_type& safe_get(int d) {
ensure(d + 1);
return (*this)[d];
}
FormalPowerSeries& cut_trailing_zeros() {
while (size() and this->back() == 0) this->pop_back();
return *this;
}
FormalPowerSeries& cut(int n) {
if (size() > n) this->resize(std::max(0, n));
return *this;
}
FormalPowerSeries cut_copy(int n) const {
FormalPowerSeries res(this->begin(), this->begin() + std::min(size(), n));
res.ensure(n);
return res;
}
FormalPowerSeries cut_copy(int l, int r) const {
if (l >= size()) return FormalPowerSeries(r - l, 0);
FormalPowerSeries res(this->begin() + l, this->begin() + std::min(size(), r));
res.ensure(r - l);
return res;
}
/* Unary Operations */
FormalPowerSeries operator+() const { return *this; }
FormalPowerSeries operator-() const {
FormalPowerSeries res = *this;
for (auto& e : res) e = -e;
return res;
}
FormalPowerSeries& operator++() { return ++safe_get(0), * this; }
FormalPowerSeries& operator--() { return --safe_get(0), * this; }
FormalPowerSeries operator++(int) {
FormalPowerSeries res = *this;
++(*this);
return res;
}
FormalPowerSeries operator--(int) {
FormalPowerSeries res = *this;
--(*this);
return res;
}
/* Binary Operations With Constant */
FormalPowerSeries& operator+=(const value_type& x) { return safe_get(0) += x, *this; }
FormalPowerSeries& operator-=(const value_type& x) { return safe_get(0) -= x, *this; }
FormalPowerSeries& operator*=(const value_type& x) {
for (auto& e : *this) e *= x;
return *this;
}
FormalPowerSeries& operator/=(const value_type& x) { return *this *= x.inv(); }
friend FormalPowerSeries operator+(FormalPowerSeries f, const value_type& x) { f += x; return f; }
friend FormalPowerSeries operator+(const value_type& x, FormalPowerSeries f) { f += x; return f; }
friend FormalPowerSeries operator-(FormalPowerSeries f, const value_type& x) { f -= x; return f; }
friend FormalPowerSeries operator-(const value_type& x, FormalPowerSeries f) { f -= x; return -f; }
friend FormalPowerSeries operator*(FormalPowerSeries f, const value_type& x) { f *= x; return f; }
friend FormalPowerSeries operator*(const value_type& x, FormalPowerSeries f) { f *= x; return f; }
friend FormalPowerSeries operator/(FormalPowerSeries f, const value_type& x) { f /= x; return f; }
/* Binary Operations With Formal Power Series */
FormalPowerSeries& operator+=(const FormalPowerSeries& g) {
const int n = g.size();
ensure(n);
for (int i = 0; i < n; ++i) (*this)[i] += g[i];
return *this;
}
FormalPowerSeries& operator-=(const FormalPowerSeries& g) {
const int n = g.size();
ensure(n);
for (int i = 0; i < n; ++i) (*this)[i] -= g[i];
return *this;
}
FormalPowerSeries& operator*=(const FormalPowerSeries& g) { return *this = *this * g; }
FormalPowerSeries& operator/=(const FormalPowerSeries& g) { return *this = *this / g; }
FormalPowerSeries& operator%=(const FormalPowerSeries& g) { return *this = *this % g; }
friend FormalPowerSeries operator+(FormalPowerSeries f, const FormalPowerSeries& g) { f += g; return f; }
friend FormalPowerSeries operator-(FormalPowerSeries f, const FormalPowerSeries& g) { f -= g; return f; }
friend FormalPowerSeries operator*(const FormalPowerSeries& f, const FormalPowerSeries& g) {
const int siz_f = f.size(), siz_g = g.size();
if (siz_f < siz_g) return g * f;
if (std::min(siz_f, siz_g) <= 60) return atcoder::convolution(f, g);
const int deg = siz_f + siz_g - 2;
int fpow2 = 1;
while ((fpow2 << 1) <= deg) fpow2 <<= 1;
if (const int dif = deg - fpow2 + 1; dif <= 10) {
FormalPowerSeries h = atcoder::convolution(std::vector<mint>(f.begin(), f.end() - dif), g);
h.resize(h.size() + dif);
for (int i = siz_f - dif; i < siz_f; ++i) for (int j = 0; j < siz_g; ++j) {
h[i + j] += f[i] * g[j];
}
return h;
}
return atcoder::convolution(f, g);
}
friend FormalPowerSeries operator/(FormalPowerSeries f, FormalPowerSeries g) {
if (f.size() < 60) return FPSNaive<mint>(f).div_mod(g).first;
f.cut_trailing_zeros(), g.cut_trailing_zeros();
const int fd = f.deg(), gd = g.deg();
assert(gd >= 0);
if (fd < gd) return {};
if (gd == 0) {
f /= g[0];
return f;
}
std::reverse(f.begin(), f.end()), std::reverse(g.begin(), g.end());
const int qd = fd - gd;
f.cut(qd + 1);
FormalPowerSeries q = f * g.inv(qd + 1);
q.cut(qd + 1);
std::reverse(q.begin(), q.end());
return q;
}
friend FormalPowerSeries operator%(const FormalPowerSeries& f, const FormalPowerSeries& g) { return f.div_mod(g).second; }
std::pair<FormalPowerSeries, FormalPowerSeries> div_mod(const FormalPowerSeries& g) const {
if (size() < 60) {
auto [q, r] = FPSNaive<mint>(*this).div_mod(g);
return { q, r };
}
FormalPowerSeries q = *this / g, r = *this - g * q;
r.cut_trailing_zeros();
return { q, r };
}
/* Shift Operations */
FormalPowerSeries& operator<<=(const int shamt) {
return this->insert(this->begin(), shamt, 0), * this;
}
FormalPowerSeries& operator>>=(const int shamt) {
return this->erase(this->begin(), this->begin() + std::min(shamt, size())), * this;
}
friend FormalPowerSeries operator<<(FormalPowerSeries f, const int shamt) { f <<= shamt; return f; }
friend FormalPowerSeries operator>>(FormalPowerSeries f, const int shamt) { f >>= shamt; return f; }
/* Compare */
friend bool operator==(const FormalPowerSeries& f, const FormalPowerSeries& g) {
const int n = f.size(), m = g.size();
if (n < m) return g == f;
for (int i = 0; i < m; ++i) if (f[i] != g[i]) return false;
for (int i = m; i < n; ++i) if (f[i] != 0) return false;
return true;
}
friend bool operator!=(const FormalPowerSeries& f, const FormalPowerSeries& g) { return not (f == g); }
/* Other Operations */
FormalPowerSeries& diff_inplace() {
const int n = size();
for (int i = 1; i < n; ++i) (*this)[i - 1] = (*this)[i] * i;
return (*this)[n - 1] = 0, *this;
}
FormalPowerSeries diff() const {
FormalPowerSeries res = *this;
res.diff_inplace();
return res;
}
FormalPowerSeries& intg_inplace() {
const int n = size();
inv_mods<value_type> invs(n);
this->resize(n + 1);
for (int i = n; i > 0; --i) (*this)[i] = (*this)[i - 1] * invs[i];
return (*this)[0] = 0, *this;
}
FormalPowerSeries intg() const {
FormalPowerSeries res = *this;
res.intg_inplace();
return res;
}
FormalPowerSeries& inv_inplace(int n = -1) { return *this = inv(n); }
// reference: https://opt-cp.com/fps-fast-algorithms/
FormalPowerSeries inv(int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).inv();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return inv_sparse(std::move(*sp_f), n);
FormalPowerSeries f_fft, g_fft;
FormalPowerSeries g{ (*this)[0].inv() };
for (int k = 1; k < n; k *= 2) {
f_fft = cut_copy(2 * k), g_fft = g.cut_copy(2 * k);
atcoder::internal::butterfly(f_fft);
atcoder::internal::butterfly(g_fft);
update_inv(k, f_fft, g_fft, g);
}
g.resize(n);
return g;
}
FormalPowerSeries& log_inplace(int n = -1) { return *this = log(n); }
FormalPowerSeries log(int n = -1) const {
assert(safe_get(0) == 1);
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).log();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return log_sparse(std::move(*sp_f), n);
FormalPowerSeries res = inv(n) * diff();
res.resize(n - 1);
return res.intg();
}
FormalPowerSeries& exp_inplace(int n = -1) { return *this = exp(n); }
// https://arxiv.org/pdf/1301.5804.pdf
FormalPowerSeries exp(int n = -1) const {
assert(safe_get(0) == 0);
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).exp();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return exp_sparse(std::move(*sp_f), n);
// h = *this
// f = exp(h) mod x ^ k
// g = f^{-1} mod x ^ k
FormalPowerSeries dh = diff();
FormalPowerSeries f{ 1 }, f_fft;
FormalPowerSeries g{ 1 }, g_fft;
for (int k = 1; k < n; k *= 2) {
f_fft = f.cut_copy(2 * k), atcoder::internal::butterfly(f_fft);
if (k > 1) update_inv(k / 2, f_fft, g_fft, g);
FormalPowerSeries t = f.cut_copy(k);
t.diff_inplace();
{
FormalPowerSeries r = dh.cut_copy(k);
r.back() = 0;
atcoder::internal::butterfly(r);
for (int i = 0; i < k; ++i) r[i] *= f_fft[i];
atcoder::internal::butterfly_inv(r);
r /= -k;
t += r;
t <<= 1, t[0] = t[k], t.pop_back();
}
t.resize(2 * k);
atcoder::internal::butterfly(t);
g_fft = g.cut_copy(2 * k);
atcoder::internal::butterfly(g_fft);
for (int i = 0; i < 2 * k; ++i) t[i] *= g_fft[i];
atcoder::internal::butterfly_inv(t);
t.resize(k);
t /= 2 * k;
FormalPowerSeries v = cut_copy(2 * k) >>= k;
t <<= k - 1;
t.intg_inplace();
for (int i = 0; i < k; ++i) v[i] -= t[k + i];
v.resize(2 * k);
atcoder::internal::butterfly(v);
for (int i = 0; i < 2 * k; ++i) v[i] *= f_fft[i];
atcoder::internal::butterfly_inv(v);
v.resize(k);
v /= 2 * k;
f.resize(2 * k);
for (int i = 0; i < k; ++i) f[k + i] = v[i];
}
f.cut(n);
return f;
}
FormalPowerSeries& pow_inplace(long long k, int n = -1) { return *this = pow(k, n); }
FormalPowerSeries pow(const long long k, int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).pow(k);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return pow_sparse(std::move(*sp_f), k, n);
if (k == 0) {
FormalPowerSeries f{ 1 };
f.resize(n);
return f;
}
int tlz = 0;
while (tlz < size() and (*this)[tlz] == 0) ++tlz;
if (tlz == size() or tlz > (n - 1) / k) return FormalPowerSeries(n, 0);
const int m = n - tlz * k;
FormalPowerSeries f = *this >> tlz;
value_type base = f[0];
return ((((f /= base).log(m) *= k).exp(m) *= base.pow(k)) <<= (tlz * k));
}
std::optional<FormalPowerSeries> safe_sqrt(int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).safe_sqrt();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return safe_sqrt_sparse(std::move(*sp_f), n);
int tlz = 0;
while (tlz < size() and (*this)[tlz] == 0) ++tlz;
if (tlz == size()) return FormalPowerSeries(n, 0);
if (tlz & 1) return std::nullopt;
const int m = n - tlz / 2;
FormalPowerSeries h(this->begin() + tlz, this->end());
auto q0 = ::safe_sqrt(h[0]);
if (not q0.has_value()) return std::nullopt;
FormalPowerSeries f{ *q0 }, f_fft, g{ q0->inv() }, g_fft;
for (int k = 1; k < m; k *= 2) {
f_fft = f.cut_copy(2 * k), atcoder::internal::butterfly(f_fft);
if (k > 1) update_inv(k / 2, f_fft, g_fft, g);
g_fft = g.cut_copy(2 * k);
atcoder::internal::butterfly(g_fft);
FormalPowerSeries h_fft = h.cut_copy(2 * k);
atcoder::internal::butterfly(h_fft);
for (int i = 0; i < 2 * k; ++i) h_fft[i] = (h_fft[i] - f_fft[i] * f_fft[i]) * g_fft[i];
atcoder::internal::butterfly_inv(h_fft);
f.resize(2 * k);
const value_type iz = value_type(4 * k).inv();
for (int i = 0; i < k; ++i) f[k + i] = h_fft[k + i] * iz;
}
f.resize(m), f <<= (tlz / 2);
return f;
}
FormalPowerSeries& sqrt_inplace(int n = -1) { return *this = sqrt(n); }
FormalPowerSeries sqrt(int n = -1) const {
return *safe_sqrt(n);
}
value_type eval(value_type x) const {
value_type y = 0;
for (int i = size() - 1; i >= 0; --i) y = y * x + (*this)[i];
return y;
}
static FormalPowerSeries prod(const std::vector<FormalPowerSeries>& fs) {
if (fs.empty()) return { 1 };
std::deque<FormalPowerSeries> dq(fs.begin(), fs.end());
std::sort(dq.begin(), dq.end(), [](auto& f, auto& g) { return f.size() < g.size(); });
while (dq.size() >= 2) {
dq.push_back(dq[0] * dq[1]);
dq.pop_front();
dq.pop_front();
}
return dq.front();
}
std::optional<std::vector<std::pair<int, value_type>>> sparse_fps_format(int max_size) const {
std::vector<std::pair<int, value_type>> res;
for (int i = 0; i <= deg() and int(res.size()) <= max_size; ++i) if (value_type v = (*this)[i]; v != 0) res.emplace_back(i, v);
if (int(res.size()) > max_size) return std::nullopt;
return res;
}
private:
static void update_inv(const int k, FormalPowerSeries& f_fft, FormalPowerSeries& g_fft, FormalPowerSeries& g) {
FormalPowerSeries fg(2 * k);
for (int i = 0; i < 2 * k; ++i) fg[i] = f_fft[i] * g_fft[i];
atcoder::internal::butterfly_inv(fg);
fg >>= k, fg.resize(2 * k);
atcoder::internal::butterfly(fg);
for (int i = 0; i < 2 * k; ++i) fg[i] *= g_fft[i];
atcoder::internal::butterfly_inv(fg);
const value_type iz = value_type(2 * k).inv(), c = -iz * iz;
g.resize(2 * k);
for (int i = 0; i < k; ++i) g[k + i] = fg[i] * c;
}
static FormalPowerSeries div_fps_sparse(const FormalPowerSeries& f, const std::vector<std::pair<int, value_type>>& g, int n) {
const int siz = g.size();
assert(siz and g[0].first == 0);
const value_type inv_g0 = g[0].second.inv();
FormalPowerSeries h(n);
for (int i = 0; i < n; ++i) {
value_type v = f.safe_get(i);
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, gj] = g[idx];
if (j > i) break;
v -= gj * h[i - j];
}
h[i] = v * inv_g0;
}
return h;
}
static FormalPowerSeries inv_sparse(const std::vector<std::pair<int, value_type>>& g, const int n) {
return div_fps_sparse(FormalPowerSeries{ 1 }, g, n);
}
static FormalPowerSeries exp_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
assert(not siz or f[0].first != 0);
FormalPowerSeries g(n);
g[0] = 1;
inv_mods<value_type> invs(n);
for (int i = 1; i < n; ++i) {
value_type v = 0;
for (const auto& [j, fj] : f) {
if (j > i) break;
v += j * fj * g[i - j];
}
v *= invs[i];
g[i] = v;
}
return g;
}
static FormalPowerSeries log_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
assert(siz and f[0].first == 0 and f[0].second == 1);
FormalPowerSeries g(n);
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, fj] = f[idx];
if (j >= n) break;
g[j] = j * fj;
}
inv_mods<value_type> invs(n);
for (int i = 1; i < n; ++i) {
value_type v = g[i];
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, fj] = f[idx];
if (j > i) break;
v -= fj * g[i - j] * (i - j);
}
v *= invs[i];
g[i] = v;
}
return g;
}
static FormalPowerSeries pow_sparse(const std::vector<std::pair<int, value_type>>& f, const long long k, const int n) {
if (k == 0) {
FormalPowerSeries res(n, 0);
res[0] = 1;
return res;
}
const int siz = f.size();
if (not siz) return FormalPowerSeries(n, 0);
const int p = f[0].first;
if (p > (n - 1) / k) return FormalPowerSeries(n, 0);
const value_type inv_f0 = f[0].second.inv();
const int lz = p * k;
FormalPowerSeries g(n);
g[lz] = f[0].second.pow(k);
inv_mods<value_type> invs(n);
for (int i = 1; lz + i < n; ++i) {
value_type v = 0;
for (int idx = 1; idx < siz; ++idx) {
auto [j, fj] = f[idx];
j -= p;
if (j > i) break;
v += fj * g[lz + i - j] * (value_type(k) * j - (i - j));
}
v *= invs[i] * inv_f0;
g[lz + i] = v;
}
return g;
}
static std::optional<FormalPowerSeries> safe_sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
if (not siz) return FormalPowerSeries(n, 0);
const int p = f[0].first;
if (p % 2 == 1) return std::nullopt;
if (p / 2 >= n) return FormalPowerSeries(n, 0);
const value_type inv_f0 = f[0].second.inv();
const int lz = p / 2;
FormalPowerSeries g(n);
auto opt_g0 = ::safe_sqrt(f[0].second);
if (not opt_g0.has_value()) return std::nullopt;
g[lz] = *opt_g0;
value_type k = mint(2).inv();
inv_mods<value_type> invs(n);
for (int i = 1; lz + i < n; ++i) {
value_type v = 0;
for (int idx = 1; idx < siz; ++idx) {
auto [j, fj] = f[idx];
j -= p;
if (j > i) break;
v += fj * g[lz + i - j] * (k * j - (i - j));
}
v *= invs[i] * inv_f0;
g[lz + i] = v;
}
return g;
}
static FormalPowerSeries sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
return *safe_sqrt(f, n);
}
};
} // namespace suisen
template <typename mint>
suisen::FormalPowerSeries<mint> sqrt(suisen::FormalPowerSeries<mint> a) {
return a.sqrt();
}
template <typename mint>
suisen::FormalPowerSeries<mint> log(suisen::FormalPowerSeries<mint> a) {
return a.log();
}
template <typename mint>
suisen::FormalPowerSeries<mint> exp(suisen::FormalPowerSeries<mint> a) {
return a.exp();
}
template <typename mint, typename T>
suisen::FormalPowerSeries<mint> pow(suisen::FormalPowerSeries<mint> a, T b) {
return a.pow(b);
}
template <typename mint>
suisen::FormalPowerSeries<mint> inv(suisen::FormalPowerSeries<mint> a) {
return a.inv();
}
using fps = FormalPowerSeries<mint>;
int main() {
int t;
read(t);
LOOP(t) {
mint a;
int n;
read(a, n);
int B = floor_kth_root(n, 2);
// B * B + r;
mint res = 0;
fps f(B);
REP(r, B) {
f[r] = a.pow(r * r);
}
vector<mint> xs(B);
REP(q, B) {
xs[q] = a.pow(2 * q * B);
}
vector<mint> ys = multi_point_eval(f, xs);
REP(q, B) {
res += a.pow((long long) q * q * B * B) * ys[q];
}
REP(i, B * B, n) {
res += a.pow((long long) i * i);
}
print(res);
}
return 0;
}