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#include "tools/extended_lucas.hpp"
It precomputes ${}_n C_r \pmod{M}$ for any $n$, any $r$ and a small fixed $M$.
tools::extended_lucas<T> cache();
It precomputes ${}_n \mathrm{C}_r \pmod{M}$ for any $n$ and $r$ where $M$ is T::mod()
.
<T>
is atcoder::static_modint
or atcoder::dynamic_modint
M cache.fact(long long n);
It returns $n! \pmod{M}$.
M cache.binomial(long long n, long long r);
It returns $\dbinom{n}{r} \pmod{M}$. Note that $\dbinom{-n}{r} = (-1)^r \dbinom{n + r - 1}{r}$.
M cache.combination(long long n, long long r);
It returns
\[\begin{align*} \left\{\begin{array}{ll} {}_n \mathrm{C}_r \pmod{M} & \text{(if $0 \leq r \leq n$)}\\ 0 \pmod{M} & \text{(otherwise)} \end{array}\right.& \end{align*}\]M cache.permutation(long long n, long long r);
It returns
\[\begin{align*} \left\{\begin{array}{ll} {}_n \mathrm{P}_r \pmod{M} & \text{(if $0 \leq r \leq n$)}\\ 0 \pmod{M} & \text{(otherwise)} \end{array}\right.& \end{align*}\]M cache.combination_with_repetition(long long n, long long r);
It returns
\[\begin{align*} \left\{\begin{array}{ll} {}_n \mathrm{H}_r \pmod{M} & \text{(if $n \geq 0$ and $r \geq 0$)}\\ 0 \pmod{M} & \text{(otherwise)} \end{array}\right.& \end{align*}\]#ifndef TOOLS_EXTENDED_LUCAS_HPP
#define TOOLS_EXTENDED_LUCAS_HPP
#include <vector>
#include <cassert>
#include <utility>
#include <iterator>
#include "tools/int128_t.hpp"
#include "tools/prime_factorization.hpp"
#include "tools/run_length.hpp"
#include "tools/garner.hpp"
// Source: https://hitonanode.github.io/cplib-cpp/number/combination.hpp.html
// License: MIT
// Author: hitonanode
// MIT License
//
// Copyright (c) 2019 Ryotaro Sato
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in all
// copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
// SOFTWARE.
namespace tools {
template <class M>
class extended_lucas {
struct combination_prime_pow {
int p, q, m;
::std::vector<int> fac, invfac, ppow;
long long ej(long long n) const {
long long ret = 0;
while (n) ret += n, n /= this->p;
return ret;
}
combination_prime_pow(const int p_, const int q_) : p(p_), q(q_), m(1), ppow{1} {
for (int t = 0; t < this->q; ++t) this->m *= this->p, this->ppow.push_back(this->m);
this->fac.assign(this->m, 1);
this->invfac.assign(this->m, 1);
for (int i = 1; i < this->m; ++i) this->fac[i] = static_cast<long long>(this->fac[i - 1]) * (i % this->p ? i : 1) % this->m;
this->invfac[this->m - 1] = this->fac[this->m - 1]; // Same as Wilson's theorem
assert(1LL * this->fac.back() * this->invfac.back() % this->m == 1);
for (int i = this->m - 1; i; --i) this->invfac[i - 1] = static_cast<long long>(this->invfac[i]) * (i % this->p ? i : 1) % this->m;
}
int fact(const long long n) const {
assert(n >= 0);
const auto q0 = this->ej(n / this->p);
return q0 < this->q ? static_cast<long long>(this->fac[n]) * this->ppow[q0] % this->m : 0;
}
int combination(long long n, long long r) const {
assert(0 <= r && r <= n);
if (this->p == 2 && this->q == 1) return !((~n) & r); // Lucas
long long k = n - r;
const long long e0 = this->ej(n / this->p) - this->ej(r / this->p) - this->ej(k / this->p);
if (e0 >= q) return 0;
long long ret = this->ppow[e0];
if (this->q == 1) { // Lucas
while (n) {
ret = ::tools::int128_t(ret) * this->fac[n % this->p] * this->invfac[r % this->p] * this->invfac[k % this->p] % this->p;
n /= this->p, r /= this->p, k /= this->p;
}
return static_cast<int>(ret);
} else {
if ((p > 2 || q < 3) && (this->ej(n / this->m) - this->ej(r / this->m) - this->ej(k / this->m)) & 1) ret = this->m - ret;
while (n) {
ret = ::tools::int128_t(ret) * this->fac[n % this->m] * this->invfac[r % this->m] * this->invfac[k % this->m] % this->m;
n /= this->p, r /= this->p, k /= this->p;
}
return static_cast<int>(ret);
}
}
};
::std::vector<combination_prime_pow> m_cpps;
public:
extended_lucas() {
const auto prime_factors = ::tools::prime_factorization(M::mod());
::std::vector<::std::pair<int, int>> distinct_prime_factors;
::tools::run_length(prime_factors.begin(), prime_factors.end(), ::std::back_inserter(distinct_prime_factors));
for (const auto& [p, q] : distinct_prime_factors) {
this->m_cpps.emplace_back(p, q);
}
}
M fact(const long long n) const {
assert(n >= 0);
::std::vector<::std::pair<int, int>> rs;
for (const auto& cpp : this->m_cpps) rs.emplace_back(cpp.fact(n), cpp.m);
return ::tools::garner<M>(rs.begin(), rs.end()).first;
}
M binomial(const long long n, const long long r) const {
if (r < 0) return M::raw(0);
if (0 <= n && n < r) return M::raw(0);
if (n < 0) return M((r & 1) ? -1 : 1) * this->binomial(-n + r - 1, r);
::std::vector<::std::pair<int, int>> rs;
for (const auto& cpp : this->m_cpps) rs.emplace_back(cpp.combination(n, r), cpp.m);
return ::tools::garner<M>(rs.begin(), rs.end()).first;
}
M combination(const long long n, const long long r) const {
if (!(0 <= r && r <= n)) return M::raw(0);
return this->binomial(n, r);
}
M permutation(const long long n, const long long r) const {
if (!(0 <= r && r <= n)) return M::raw(0);
return this->binomial(n, r) * this->fact(r);
}
M combination_with_repetition(const long long n, const long long r) const {
if (n < 0 || r < 0) return M::raw(0);
return this->binomial(n + r - 1, r);
}
};
}
#endif
#line 1 "tools/extended_lucas.hpp"
#include <vector>
#include <cassert>
#include <utility>
#include <iterator>
#line 1 "tools/int128_t.hpp"
#line 1 "tools/detail/int128_t.hpp"
#include <algorithm>
#line 6 "tools/detail/int128_t.hpp"
#include <cstddef>
#include <cstdint>
#include <functional>
#include <iostream>
#include <limits>
#include <string>
#include <string_view>
#line 1 "tools/abs.hpp"
namespace tools {
constexpr float abs(const float x) {
return x < 0 ? -x : x;
}
constexpr double abs(const double x) {
return x < 0 ? -x : x;
}
constexpr long double abs(const long double x) {
return x < 0 ? -x : x;
}
constexpr int abs(const int x) {
return x < 0 ? -x : x;
}
constexpr long abs(const long x) {
return x < 0 ? -x : x;
}
constexpr long long abs(const long long x) {
return x < 0 ? -x : x;
}
constexpr unsigned int abs(const unsigned int x) {
return x;
}
constexpr unsigned long abs(const unsigned long x) {
return x;
}
constexpr unsigned long long abs(const unsigned long long x) {
return x;
}
}
#line 1 "tools/bit_ceil.hpp"
#include <bit>
#line 6 "tools/bit_ceil.hpp"
#include <type_traits>
#line 1 "tools/is_integral.hpp"
#line 5 "tools/is_integral.hpp"
namespace tools {
template <typename T>
struct is_integral : ::std::is_integral<T> {};
template <typename T>
inline constexpr bool is_integral_v = ::tools::is_integral<T>::value;
}
#line 1 "tools/is_signed.hpp"
#line 5 "tools/is_signed.hpp"
namespace tools {
template <typename T>
struct is_signed : ::std::is_signed<T> {};
template <typename T>
inline constexpr bool is_signed_v = ::tools::is_signed<T>::value;
}
#line 1 "tools/make_unsigned.hpp"
#line 5 "tools/make_unsigned.hpp"
namespace tools {
template <typename T>
struct make_unsigned : ::std::make_unsigned<T> {};
template <typename T>
using make_unsigned_t = typename ::tools::make_unsigned<T>::type;
}
#line 10 "tools/bit_ceil.hpp"
namespace tools {
template <typename T>
constexpr T bit_ceil(T) noexcept;
template <typename T>
constexpr T bit_ceil(const T x) noexcept {
static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
if constexpr (::tools::is_signed_v<T>) {
assert(x >= 0);
return ::tools::bit_ceil<::tools::make_unsigned_t<T>>(x);
} else {
return ::std::bit_ceil(x);
}
}
}
#line 1 "tools/bit_floor.hpp"
#line 10 "tools/bit_floor.hpp"
namespace tools {
template <typename T>
constexpr T bit_floor(T) noexcept;
template <typename T>
constexpr T bit_floor(const T x) noexcept {
static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
if constexpr (::tools::is_signed_v<T>) {
assert(x >= 0);
return ::tools::bit_floor<::tools::make_unsigned_t<T>>(x);
} else {
return ::std::bit_floor(x);
}
}
}
#line 1 "tools/bit_width.hpp"
#line 10 "tools/bit_width.hpp"
namespace tools {
template <typename T>
constexpr int bit_width(T) noexcept;
template <typename T>
constexpr int bit_width(const T x) noexcept {
static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
if constexpr (::tools::is_signed_v<T>) {
assert(x >= 0);
return ::tools::bit_width<::tools::make_unsigned_t<T>>(x);
} else {
return ::std::bit_width(x);
}
}
}
#line 1 "tools/countr_zero.hpp"
#line 12 "tools/countr_zero.hpp"
namespace tools {
template <typename T>
constexpr int countr_zero(const T x) noexcept {
static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
if constexpr (::tools::is_signed_v<T>) {
assert(x >= 0);
return ::std::min(::tools::countr_zero<::tools::make_unsigned_t<T>>(x), ::std::numeric_limits<T>::digits);
} else {
return ::std::countr_zero(x);
}
}
}
#line 1 "tools/is_unsigned.hpp"
#line 5 "tools/is_unsigned.hpp"
namespace tools {
template <typename T>
struct is_unsigned : ::std::is_unsigned<T> {};
template <typename T>
inline constexpr bool is_unsigned_v = ::tools::is_unsigned<T>::value;
}
#line 1 "tools/hash_combine.hpp"
#line 6 "tools/hash_combine.hpp"
// Source: https://github.com/google/cityhash/blob/f5dc54147fcce12cefd16548c8e760d68ac04226/src/city.h
// License: MIT
// Author: Google Inc.
// Copyright (c) 2011 Google, Inc.
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE.
namespace tools {
template <typename T>
void hash_combine(::std::size_t& seed, const T& v) {
static const ::std::hash<T> hasher;
static constexpr ::std::size_t k_mul = 0x9ddfea08eb382d69ULL;
::std::size_t a = (hasher(v) ^ seed) * k_mul;
a ^= (a >> 47);
::std::size_t b = (seed ^ a) * k_mul;
b ^= (b >> 47);
seed = b * k_mul;
}
}
#line 1 "tools/make_signed.hpp"
#line 5 "tools/make_signed.hpp"
namespace tools {
template <typename T>
struct make_signed : ::std::make_signed<T> {};
template <typename T>
using make_signed_t = typename ::tools::make_signed<T>::type;
}
#line 1 "tools/now.hpp"
#include <chrono>
namespace tools {
inline long long now() {
return ::std::chrono::duration_cast<::std::chrono::nanoseconds>(::std::chrono::high_resolution_clock::now().time_since_epoch()).count();
}
}
#line 25 "tools/detail/int128_t.hpp"
namespace tools {
using uint128_t = unsigned __int128;
using int128_t = __int128;
namespace detail {
namespace int128_t {
constexpr ::tools::uint128_t parse_unsigned(const ::std::string_view s) noexcept {
assert(!s.empty());
::tools::uint128_t x = 0;
::std::size_t i = s[0] == '+';
if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
for (i += 2; i < s.size(); ++i) {
assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
x <<= 4;
if ('0' <= s[i] && s[i] <= '9') {
x |= s[i] - '0';
} else if ('a' <= s[i] && s[i] <= 'f') {
x |= s[i] - 'a' + 10;
} else {
x |= s[i] - 'A' + 10;
}
}
} else {
for (; i < s.size(); ++i) {
assert('0' <= s[i] && s[i] <= '9');
x *= 10;
x += s[i] - '0';
}
}
return x;
}
constexpr ::tools::int128_t parse_signed(const ::std::string_view s) noexcept {
assert(!s.empty());
::tools::int128_t x = 0;
if (s[0] == '-') {
::std::size_t i = 1;
if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
for (i += 2; i < s.size(); ++i) {
assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
x *= 16;
if ('0' <= s[i] && s[i] <= '9') {
x -= s[i] - '0';
} else if ('a' <= s[i] && s[i] <= 'f') {
x -= s[i] - 'a' + 10;
} else {
x -= s[i] - 'A' + 10;
}
}
} else {
for (; i < s.size(); ++i) {
assert('0' <= s[i] && s[i] <= '9');
x *= 10;
x -= s[i] - '0';
}
}
} else {
::std::size_t i = s[0] == '+';
if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
for (i += 2; i < s.size(); ++i) {
assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
x <<= 4;
if ('0' <= s[i] && s[i] <= '9') {
x |= s[i] - '0';
} else if ('a' <= s[i] && s[i] <= 'f') {
x |= s[i] - 'a' + 10;
} else {
x |= s[i] - 'A' + 10;
}
}
} else {
for (; i < s.size(); ++i) {
assert('0' <= s[i] && s[i] <= '9');
x *= 10;
x += s[i] - '0';
}
}
}
return x;
}
}
}
constexpr ::tools::uint128_t abs(const ::tools::uint128_t& x) noexcept {
return x;
}
constexpr ::tools::int128_t abs(const ::tools::int128_t& x) {
return x >= 0 ? x : -x;
}
}
#define UINT128_C(c) ::tools::detail::int128_t::parse_unsigned(#c)
#define INT128_C(c) ::tools::detail::int128_t::parse_signed(#c)
inline ::std::istream& operator>>(::std::istream& is, ::tools::uint128_t& x) {
::std::string s;
is >> s;
x = ::tools::detail::int128_t::parse_unsigned(s);
return is;
}
inline ::std::istream& operator>>(::std::istream& is, ::tools::int128_t& x) {
::std::string s;
is >> s;
x = ::tools::detail::int128_t::parse_signed(s);
return is;
}
inline ::std::ostream& operator<<(::std::ostream& os, ::tools::uint128_t x) {
::std::string s;
if (x > 0) {
while (x > 0) {
s.push_back('0' + x % 10);
x /= 10;
}
} else {
s.push_back('0');
}
::std::ranges::reverse(s);
return os << s;
}
inline ::std::ostream& operator<<(::std::ostream& os, ::tools::int128_t x) {
::std::string s;
if (x > 0) {
while (x > 0) {
s.push_back('0' + x % 10);
x /= 10;
}
} else if (x < 0) {
while (x < 0) {
s.push_back('0' + (-(x % 10)));
x /= 10;
}
s.push_back('-');
} else {
s.push_back('0');
}
::std::ranges::reverse(s);
return os << s;
}
#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
namespace std {
template <>
struct hash<::tools::uint128_t> {
::std::size_t operator()(const ::tools::uint128_t& x) const {
static const ::std::size_t seed = ::tools::now();
::std::size_t hash = seed;
::tools::hash_combine(hash, static_cast<::std::uint64_t>(x >> 64));
::tools::hash_combine(hash, static_cast<::std::uint64_t>(x & ((UINT128_C(1) << 64) - 1)));
return hash;
}
};
template <>
struct hash<::tools::int128_t> {
::std::size_t operator()(const ::tools::int128_t& x) const {
static ::std::hash<::tools::uint128_t> hasher;
return hasher(static_cast<::tools::uint128_t>(x));
}
};
}
#endif
namespace tools {
template <>
struct is_integral<::tools::int128_t> : ::std::true_type {};
template <>
struct is_integral<::tools::uint128_t> : ::std::true_type {};
template <>
struct is_integral<const ::tools::int128_t> : ::std::true_type {};
template <>
struct is_integral<const ::tools::uint128_t> : ::std::true_type {};
template <>
struct is_integral<volatile ::tools::int128_t> : ::std::true_type {};
template <>
struct is_integral<volatile ::tools::uint128_t> : ::std::true_type {};
template <>
struct is_integral<const volatile ::tools::int128_t> : ::std::true_type {};
template <>
struct is_integral<const volatile ::tools::uint128_t> : ::std::true_type {};
template <>
struct is_signed<::tools::int128_t> : ::std::true_type {};
template <>
struct is_signed<::tools::uint128_t> : ::std::false_type {};
template <>
struct is_signed<const ::tools::int128_t> : ::std::true_type {};
template <>
struct is_signed<const ::tools::uint128_t> : ::std::false_type {};
template <>
struct is_signed<volatile ::tools::int128_t> : ::std::true_type {};
template <>
struct is_signed<volatile ::tools::uint128_t> : ::std::false_type {};
template <>
struct is_signed<const volatile ::tools::int128_t> : ::std::true_type {};
template <>
struct is_signed<const volatile ::tools::uint128_t> : ::std::false_type {};
template <>
struct is_unsigned<::tools::int128_t> : ::std::false_type {};
template <>
struct is_unsigned<::tools::uint128_t> : ::std::true_type {};
template <>
struct is_unsigned<const ::tools::int128_t> : ::std::false_type {};
template <>
struct is_unsigned<const ::tools::uint128_t> : ::std::true_type {};
template <>
struct is_unsigned<volatile ::tools::int128_t> : ::std::false_type {};
template <>
struct is_unsigned<volatile ::tools::uint128_t> : ::std::true_type {};
template <>
struct is_unsigned<const volatile ::tools::int128_t> : ::std::false_type {};
template <>
struct is_unsigned<const volatile ::tools::uint128_t> : ::std::true_type {};
template <>
struct make_signed<::tools::int128_t> {
using type = ::tools::int128_t;
};
template <>
struct make_signed<::tools::uint128_t> {
using type = ::tools::int128_t;
};
template <>
struct make_signed<const ::tools::int128_t> {
using type = const ::tools::int128_t;
};
template <>
struct make_signed<const ::tools::uint128_t> {
using type = const ::tools::int128_t;
};
template <>
struct make_signed<volatile ::tools::int128_t> {
using type = volatile ::tools::int128_t;
};
template <>
struct make_signed<volatile ::tools::uint128_t> {
using type = volatile ::tools::int128_t;
};
template <>
struct make_signed<const volatile ::tools::int128_t> {
using type = const volatile ::tools::int128_t;
};
template <>
struct make_signed<const volatile ::tools::uint128_t> {
using type = const volatile ::tools::int128_t;
};
template <>
struct make_unsigned<::tools::int128_t> {
using type = ::tools::uint128_t;
};
template <>
struct make_unsigned<::tools::uint128_t> {
using type = ::tools::uint128_t;
};
template <>
struct make_unsigned<const ::tools::int128_t> {
using type = const ::tools::uint128_t;
};
template <>
struct make_unsigned<const ::tools::uint128_t> {
using type = const ::tools::uint128_t;
};
template <>
struct make_unsigned<volatile ::tools::int128_t> {
using type = volatile ::tools::uint128_t;
};
template <>
struct make_unsigned<volatile ::tools::uint128_t> {
using type = volatile ::tools::uint128_t;
};
template <>
struct make_unsigned<const volatile ::tools::int128_t> {
using type = const volatile ::tools::uint128_t;
};
template <>
struct make_unsigned<const volatile ::tools::uint128_t> {
using type = const volatile ::tools::uint128_t;
};
#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
template <>
constexpr ::tools::uint128_t bit_ceil<::tools::uint128_t>(::tools::uint128_t x) noexcept {
if (x <= 1) return 1;
--x;
x |= x >> 1;
x |= x >> 2;
x |= x >> 4;
x |= x >> 8;
x |= x >> 16;
x |= x >> 32;
x |= x >> 64;
return ++x;
}
template <>
constexpr ::tools::uint128_t bit_floor<::tools::uint128_t>(::tools::uint128_t x) noexcept {
x |= x >> 1;
x |= x >> 2;
x |= x >> 4;
x |= x >> 8;
x |= x >> 16;
x |= x >> 32;
x |= x >> 64;
return x & ~(x >> 1);
}
template <>
constexpr int bit_width<::tools::uint128_t>(::tools::uint128_t x) noexcept {
int w = 0;
if (x & UINT128_C(0xffffffffffffffff0000000000000000)) {
x >>= 64;
w += 64;
}
if (x & UINT128_C(0xffffffff00000000)) {
x >>= 32;
w += 32;
}
if (x & UINT128_C(0xffff0000)) {
x >>= 16;
w += 16;
}
if (x & UINT128_C(0xff00)) {
x >>= 8;
w += 8;
}
if (x & UINT128_C(0xf0)) {
x >>= 4;
w += 4;
}
if (x & UINT128_C(0xc)) {
x >>= 2;
w += 2;
}
if (x & UINT128_C(0x2)) {
x >>= 1;
w += 1;
}
w += x;
return w;
}
namespace detail {
namespace countr_zero {
template <::std::size_t N>
struct ntz_traits;
template <>
struct ntz_traits<128> {
using type = ::tools::uint128_t;
static constexpr int shift = 120;
static constexpr type magic = UINT128_C(0x01061438916347932a5cd9d3ead7b77f);
static constexpr int ntz_table[255] = {
128, 0, 1, -1, 2, -1, 8, -1, 3, -1, 15, -1, 9, -1, 22, -1,
4, -1, 29, -1, 16, -1, 36, -1, 10, -1, 43, -1, 23, -1, 50, -1,
5, -1, 33, -1, 30, -1, 57, -1, 17, -1, 64, -1, 37, -1, 71, -1,
11, -1, 60, -1, 44, -1, 78, -1, 24, -1, 85, -1, 51, -1, 92, -1,
-1, 6, -1, 20, -1, 34, -1, 48, 31, -1, -1, 69, 58, -1, -1, 90,
18, -1, 67, -1, 65, -1, 99, -1, 38, -1, 101, -1, 72, -1, 106, -1,
-1, 12, -1, 40, -1, 61, -1, 82, 45, -1, -1, 103, 79, -1, 113, -1,
-1, 25, -1, 74, 86, -1, -1, 116, -1, 52, -1, 108, -1, 93, -1, 120,
127, -1, -1, 7, -1, 14, -1, 21, -1, 28, -1, 35, -1, 42, -1, 49,
-1, 32, -1, 56, -1, 63, -1, 70, -1, 59, -1, 77, -1, 84, -1, 91,
-1, 19, -1, 47, -1, 68, -1, 89, -1, 66, -1, 98, -1, 100, -1, 105,
-1, 39, -1, 81, -1, 102, -1, 112, -1, 73, -1, 115, -1, 107, -1, 119,
126, -1, 13, -1, 27, -1, 41, -1, -1, 55, 62, -1, -1, 76, 83, -1,
-1, 46, -1, 88, -1, 97, -1, 104, -1, 80, -1, 111, -1, 114, -1, 118,
125, -1, 26, -1, 54, -1, 75, -1, -1, 87, 96, -1, -1, 110, -1, 117,
124, -1, 53, -1, -1, 95, 109, -1, 123, -1, 94, -1, 122, -1, 121
};
};
template <typename T>
constexpr int impl(const T x) noexcept {
using tr = ::tools::detail::countr_zero::ntz_traits<::std::numeric_limits<T>::digits>;
using type = typename tr::type;
return tr::ntz_table[static_cast<type>(tr::magic * static_cast<type>(x & -x)) >> tr::shift];
}
}
}
template <>
constexpr int countr_zero<::tools::uint128_t>(const ::tools::uint128_t x) noexcept {
return ::tools::detail::countr_zero::impl(x);
}
#endif
}
#line 5 "tools/int128_t.hpp"
#line 1 "tools/prime_factorization.hpp"
#line 6 "tools/prime_factorization.hpp"
#include <queue>
#line 9 "tools/prime_factorization.hpp"
#include <cmath>
#include <numeric>
#line 1 "tools/is_prime.hpp"
#include <array>
#line 1 "tools/prod_mod.hpp"
#line 1 "tools/uint128_t.hpp"
#line 5 "tools/uint128_t.hpp"
#line 5 "tools/prod_mod.hpp"
namespace tools {
template <typename T1, typename T2, typename T3>
constexpr T3 prod_mod(const T1 x, const T2 y, const T3 m) {
using u128 = ::tools::uint128_t;
u128 prod_mod = u128(x >= 0 ? x : -x) * u128(y >= 0 ? y : -y) % u128(m);
if ((x >= 0) ^ (y >= 0)) prod_mod = u128(m) - prod_mod;
return prod_mod;
}
}
#line 1 "tools/pow_mod.hpp"
#line 1 "tools/mod.hpp"
#line 7 "tools/mod.hpp"
namespace tools {
template <typename M, typename N> requires (
::tools::is_integral_v<M> && !::std::is_same_v<::std::remove_cv_t<M>, bool> &&
::tools::is_integral_v<N> && !::std::is_same_v<::std::remove_cv_t<N>, bool>)
constexpr ::std::common_type_t<M, N> mod(const M a, const N b) noexcept {
assert(b != 0);
using UM = ::std::make_unsigned_t<M>;
using UN = ::std::make_unsigned_t<N>;
const UM ua = a >= 0 ? a : static_cast<UM>(-(a + 1)) + 1;
const UN ub = b >= 0 ? b : static_cast<UN>(-(b + 1)) + 1;
auto r = ua % ub;
if (a < 0 && r > 0) {
r = ub - r;
}
return r;
}
}
#line 6 "tools/pow_mod.hpp"
namespace tools {
template <typename T1, typename T2, typename T3>
constexpr T3 pow_mod(const T1 x, T2 n, const T3 m) {
if (m == 1) return 0;
T3 r = 1;
T3 y = ::tools::mod(x, m);
while (n > 0) {
if ((n & 1) > 0) {
r = ::tools::prod_mod(r, y, m);
}
y = ::tools::prod_mod(y, y, m);
n /= 2;
}
return r;
}
}
#line 7 "tools/is_prime.hpp"
namespace tools {
constexpr bool is_prime(const unsigned long long n) {
constexpr ::std::array<unsigned long long, 7> bases = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
if (n <= 1) return false;
if (n == 2) return true;
if (n % 2 == 0) return false;
auto d = n - 1;
for (; d % 2 == 0; d /= 2);
for (const auto a : bases) {
if (a % n == 0) return true;
auto power = d;
auto target = ::tools::pow_mod(a, power, n);
bool is_composite = true;
if (target == 1) is_composite = false;
for (; is_composite && power != n - 1; power *= 2, target = ::tools::prod_mod(target, target, n)) {
if (target == n - 1) is_composite = false;
}
if (is_composite) {
return false;
}
}
return true;
}
}
#line 1 "tools/pow2.hpp"
#line 6 "tools/pow2.hpp"
namespace tools {
template <typename T, typename ::std::enable_if<::std::is_unsigned<T>::value, ::std::nullptr_t>::type = nullptr>
constexpr T pow2(const T x) {
return static_cast<T>(1) << x;
}
template <typename T, typename ::std::enable_if<::std::is_signed<T>::value, ::std::nullptr_t>::type = nullptr>
constexpr T pow2(const T x) {
return static_cast<T>(static_cast<typename ::std::make_unsigned<T>::type>(1) << static_cast<typename ::std::make_unsigned<T>::type>(x));
}
}
#line 1 "tools/floor_log2.hpp"
#line 6 "tools/floor_log2.hpp"
namespace tools {
template <typename T>
constexpr T floor_log2(T x) noexcept {
assert(x > 0);
return ::tools::bit_width(x) - 1;
}
}
#line 15 "tools/prime_factorization.hpp"
namespace tools {
template <typename T>
::std::vector<T> prime_factorization(T n) {
assert(1 <= n && n <= 1000000000000000000);
::std::vector<T> result;
if (n == 1) return result;
::std::queue<::std::pair<T, T>> factors({::std::pair<T, T>(n, 1)});
while (!factors.empty()) {
const T factor = factors.front().first;
const T occurrences = factors.front().second;
factors.pop();
if (::tools::is_prime(factor)) {
for (T i = 0; i < occurrences; ++i) {
result.push_back(factor);
}
} else {
const T m = ::tools::pow2((::tools::floor_log2(factor) + 1) / 8);
for (T c = 1; ; ++c) {
const auto f = [&](T& x) {
x = ::tools::prod_mod(x, x, factor);
x += c;
if (x >= factor) x -= factor;
};
T y = 2;
T r = 1;
T q = 1;
T x, g, ys;
do {
x = y;
for (T i = 0; i < r; ++i) {
f(y);
}
T k = 0;
do {
ys = y;
for (T i = 0; i < ::std::min(m, r - k); ++i) {
f(y);
q = ::tools::prod_mod(q, ::std::abs(x - y), factor);
}
g = ::std::gcd(q, factor);
k += m;
} while (k < r && g == 1);
r *= 2;
} while (g == 1);
if (g == factor) {
do {
f(ys);
g = ::std::gcd(::std::abs(x - ys), factor);
} while (g == 1);
}
if (g < factor) {
T h = factor / g;
if (h < g) ::std::swap(g, h);
T n = 1;
while (h % g == 0) {
h /= g;
++n;
}
factors.emplace(g, occurrences * n);
if (h > 1) factors.emplace(h, occurrences);
break;
}
}
}
}
::std::sort(result.begin(), result.end());
return result;
}
}
#line 1 "tools/run_length.hpp"
#line 8 "tools/run_length.hpp"
namespace tools {
template <typename InputIterator, typename OutputIterator>
void run_length(const InputIterator& begin, const InputIterator& end, OutputIterator result) {
using T = typename ::std::iterator_traits<InputIterator>::value_type;
if (begin == end) return;
::std::pair<T, ::std::size_t> prev;
for (auto [it, breaks] = ::std::make_pair(begin, false); !breaks; breaks = it == end, it = ::std::next(it, breaks ? 0 : 1)) {
bool flg1, flg2;
if (it == begin) {
flg1 = false;
flg2 = true;
} else if (it == end) {
flg1 = true;
flg2 = false;
} else if (*it != prev.first) {
flg1 = true;
flg2 = true;
} else {
flg1 = false;
flg2 = false;
}
if (flg1 || flg2) {
if (flg1) {
*result = prev;
++result;
}
if (flg2) {
prev.first = *it;
prev.second = 1;
}
} else {
++prev.second;
}
}
}
}
#line 1 "tools/garner.hpp"
#line 1 "tools/inv_mod.hpp"
#line 1 "tools/extgcd.hpp"
#include <tuple>
#line 9 "tools/extgcd.hpp"
namespace tools {
template <typename T>
::std::tuple<T, T, T> extgcd(T prev_r, T r) {
const bool prev_r_is_neg = prev_r < T(0);
const bool r_is_neg = r < T(0);
if (prev_r_is_neg) prev_r = -prev_r;
if (r_is_neg) r = -r;
#ifndef NDEBUG
const T a = prev_r;
const T b = r;
#endif
T prev_s(1);
T prev_t(0);
T s(0);
T t(1);
while (r != T(0)) {
const T q = prev_r / r;
::std::tie(prev_r, r) = ::std::make_pair(r, prev_r - q * r);
::std::tie(prev_s, s) = ::std::make_pair(s, prev_s - q * s);
::std::tie(prev_t, t) = ::std::make_pair(t, prev_t - q * t);
}
if (prev_r_is_neg) prev_s = -prev_s;
if (r_is_neg) prev_t = -prev_t;
assert(::tools::abs(prev_s) <= ::std::max(b / prev_r / T(2), T(1)));
assert(::tools::abs(prev_t) <= ::std::max(a / prev_r / T(2), T(1)));
return ::std::make_tuple(prev_s, prev_t, prev_r);
}
}
#line 7 "tools/inv_mod.hpp"
namespace tools {
template <typename T1, typename T2>
constexpr T2 inv_mod(const T1 x, const T2 m) {
const auto [x0, y0, gcd] = ::tools::extgcd(x, m);
assert(gcd == 1);
return ::tools::mod(x0, m);
}
}
#line 9 "tools/garner.hpp"
// Source: https://qiita.com/drken/items/ae02240cd1f8edfc86fd
// License: unknown
// Author: drken
namespace tools {
template <typename Iterator, typename ModType>
::std::pair<long long, long long> garner(const Iterator& begin, const Iterator& end, const ModType& mod) {
::std::vector<long long> b, m;
for (auto it = begin; it != end; ++it) {
b.push_back(::tools::mod(it->first, it->second));
m.push_back(it->second);
}
auto lcm = 1LL;
for (::std::size_t i = 0; i < b.size(); ++i) {
(lcm *= m[i]) %= mod;
}
m.push_back(mod);
::std::vector<long long> coeffs(m.size(), 1);
::std::vector<long long> constants(m.size(), 0);
for (::std::size_t k = 0; k < b.size(); ++k) {
long long t = ::tools::mod((b[k] - constants[k]) * ::tools::inv_mod(coeffs[k], m[k]), m[k]);
for (::std::size_t i = k + 1; i < m.size(); ++i) {
(constants[i] += t * coeffs[i]) %= m[i];
(coeffs[i] *= m[k]) %= m[i];
}
}
return ::std::make_pair(constants.back(), lcm);
}
template <typename M, typename Iterator>
::std::pair<M, M> garner(const Iterator& begin, const Iterator& end) {
const auto [y, z] = ::tools::garner(begin, end, M::mod());
return ::std::make_pair(M::raw(y), M::raw(z));
}
}
#line 12 "tools/extended_lucas.hpp"
// Source: https://hitonanode.github.io/cplib-cpp/number/combination.hpp.html
// License: MIT
// Author: hitonanode
// MIT License
//
// Copyright (c) 2019 Ryotaro Sato
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in all
// copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
// SOFTWARE.
namespace tools {
template <class M>
class extended_lucas {
struct combination_prime_pow {
int p, q, m;
::std::vector<int> fac, invfac, ppow;
long long ej(long long n) const {
long long ret = 0;
while (n) ret += n, n /= this->p;
return ret;
}
combination_prime_pow(const int p_, const int q_) : p(p_), q(q_), m(1), ppow{1} {
for (int t = 0; t < this->q; ++t) this->m *= this->p, this->ppow.push_back(this->m);
this->fac.assign(this->m, 1);
this->invfac.assign(this->m, 1);
for (int i = 1; i < this->m; ++i) this->fac[i] = static_cast<long long>(this->fac[i - 1]) * (i % this->p ? i : 1) % this->m;
this->invfac[this->m - 1] = this->fac[this->m - 1]; // Same as Wilson's theorem
assert(1LL * this->fac.back() * this->invfac.back() % this->m == 1);
for (int i = this->m - 1; i; --i) this->invfac[i - 1] = static_cast<long long>(this->invfac[i]) * (i % this->p ? i : 1) % this->m;
}
int fact(const long long n) const {
assert(n >= 0);
const auto q0 = this->ej(n / this->p);
return q0 < this->q ? static_cast<long long>(this->fac[n]) * this->ppow[q0] % this->m : 0;
}
int combination(long long n, long long r) const {
assert(0 <= r && r <= n);
if (this->p == 2 && this->q == 1) return !((~n) & r); // Lucas
long long k = n - r;
const long long e0 = this->ej(n / this->p) - this->ej(r / this->p) - this->ej(k / this->p);
if (e0 >= q) return 0;
long long ret = this->ppow[e0];
if (this->q == 1) { // Lucas
while (n) {
ret = ::tools::int128_t(ret) * this->fac[n % this->p] * this->invfac[r % this->p] * this->invfac[k % this->p] % this->p;
n /= this->p, r /= this->p, k /= this->p;
}
return static_cast<int>(ret);
} else {
if ((p > 2 || q < 3) && (this->ej(n / this->m) - this->ej(r / this->m) - this->ej(k / this->m)) & 1) ret = this->m - ret;
while (n) {
ret = ::tools::int128_t(ret) * this->fac[n % this->m] * this->invfac[r % this->m] * this->invfac[k % this->m] % this->m;
n /= this->p, r /= this->p, k /= this->p;
}
return static_cast<int>(ret);
}
}
};
::std::vector<combination_prime_pow> m_cpps;
public:
extended_lucas() {
const auto prime_factors = ::tools::prime_factorization(M::mod());
::std::vector<::std::pair<int, int>> distinct_prime_factors;
::tools::run_length(prime_factors.begin(), prime_factors.end(), ::std::back_inserter(distinct_prime_factors));
for (const auto& [p, q] : distinct_prime_factors) {
this->m_cpps.emplace_back(p, q);
}
}
M fact(const long long n) const {
assert(n >= 0);
::std::vector<::std::pair<int, int>> rs;
for (const auto& cpp : this->m_cpps) rs.emplace_back(cpp.fact(n), cpp.m);
return ::tools::garner<M>(rs.begin(), rs.end()).first;
}
M binomial(const long long n, const long long r) const {
if (r < 0) return M::raw(0);
if (0 <= n && n < r) return M::raw(0);
if (n < 0) return M((r & 1) ? -1 : 1) * this->binomial(-n + r - 1, r);
::std::vector<::std::pair<int, int>> rs;
for (const auto& cpp : this->m_cpps) rs.emplace_back(cpp.combination(n, r), cpp.m);
return ::tools::garner<M>(rs.begin(), rs.end()).first;
}
M combination(const long long n, const long long r) const {
if (!(0 <= r && r <= n)) return M::raw(0);
return this->binomial(n, r);
}
M permutation(const long long n, const long long r) const {
if (!(0 <= r && r <= n)) return M::raw(0);
return this->binomial(n, r) * this->fact(r);
}
M combination_with_repetition(const long long n, const long long r) const {
if (n < 0 || r < 0) return M::raw(0);
return this->binomial(n + r - 1, r);
}
};
}