proconlib
This documentation is automatically generated by competitive-verifier/competitive-verifier
tests/ord_mod/count.test.cpp
- View this file on GitHub
- Last update: 2025-02-10 20:05:30+09:00
- Problem: https://atcoder.jp/contests/abc212/tasks/abc212_g
Depends on
std::abs(x) extended for my library (tools/abs.hpp)- $2^{\left\lceil \log_2(x) \right\rceil}$ (tools/bit_ceil.hpp)
- $2^{\left\lfloor \log_2(x) \right\rfloor}$ (tools/bit_floor.hpp)
- Bit width required to represent a given integer (tools/bit_width.hpp)
- Number of trailing zeros (tools/countr_zero.hpp)
- tools/detail/int128_t.hpp
- Fixed point combinator (tools/fix.hpp)
- $\left\lfloor \log_2(x) \right\rfloor$ (tools/floor_log2.hpp)
- std::gcd(m, n) extended for my library (tools/gcd.hpp)
- Combine hash values (tools/hash_combine.hpp)
- std::is_integral extended for tools::int128_t and tools::uint128_t (tools/is_integral.hpp)
- Check whether T is a monoid (tools/is_monoid.hpp)
- Miller-Rabin primality test (tools/is_prime.hpp)
- std::is_signed extended for tools::int128_t and tools::uint128_t (tools/is_signed.hpp)
- std::is_unsigned extended for tools::int128_t and tools::uint128_t (tools/is_unsigned.hpp)
- std::make_signed extended for tools::int128_t and tools::uint128_t (tools/make_signed.hpp)
- std::make_unsigned extended for tools::int128_t and tools::uint128_t (tools/make_unsigned.hpp)
- Minimum non-negative reminder (tools/mod.hpp)
- Typical monoids (tools/monoid.hpp)
- The number of nanoseconds that have elapsed since epoch (tools/now.hpp)
- $\mathrm{ord}(x)$ for $x \in (\mathbb{Z}/p\mathbb{Z})^\times$ (tools/ord_mod.hpp)
- $b^n$ under a given monoid, and std::pow(b, n) extended for my library (tools/pow.hpp)
- $2^x$ (tools/pow2.hpp)
- $x^y \pmod{M}$ (tools/pow_mod.hpp)
- Pollard's rho algorithm (tools/prime_factorization.hpp)
- $x \cdot y \pmod{M}$ (tools/prod_mod.hpp)
- $x^2$ under a given monoid (tools/square.hpp)
- 128-bit unsigned integer (tools/uint128_t.hpp)
Code
// competitive-verifier: PROBLEM https://atcoder.jp/contests/abc212/tasks/abc212_g
// competitive-verifier: IGNORE
#include <iostream>
#include "atcoder/modint.hpp"
#include "tools/ord_mod.hpp"
using mint = atcoder::modint998244353;
using ll = long long;
int main() {
std::cin.tie(nullptr);
std::ios_base::sync_with_stdio(false);
ll P;
std::cin >> P;
auto answer = mint::raw(1);
for (const auto& [ord, n] : tools::ord_mod(P).count()) {
answer += mint(ord) * mint(n);
}
std::cout << answer.val() << '\n';
return 0;
}
#line 1 "tests/ord_mod/count.test.cpp"
// competitive-verifier: PROBLEM https://atcoder.jp/contests/abc212/tasks/abc212_g
// competitive-verifier: IGNORE
#include <iostream>
#line 1 "lib/ac-library/atcoder/modint.hpp"
#include <cassert>
#include <numeric>
#include <type_traits>
#ifdef _MSC_VER
#include <intrin.h>
#endif
#line 1 "lib/ac-library/atcoder/internal_math.hpp"
#include <utility>
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
x %= m;
if (x < 0) x += m;
return x;
}
// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
unsigned int _m;
unsigned long long im;
// @param m `1 <= m`
explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
// @return m
unsigned int umod() const { return _m; }
// @param a `0 <= a < m`
// @param b `0 <= b < m`
// @return `a * b % m`
unsigned int mul(unsigned int a, unsigned int b) const {
// [1] m = 1
// a = b = im = 0, so okay
// [2] m >= 2
// im = ceil(2^64 / m)
// -> im * m = 2^64 + r (0 <= r < m)
// let z = a*b = c*m + d (0 <= c, d < m)
// a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
// c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
// ((ab * im) >> 64) == c or c + 1
unsigned long long z = a;
z *= b;
#ifdef _MSC_VER
unsigned long long x;
_umul128(z, im, &x);
#else
unsigned long long x =
(unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
unsigned long long y = x * _m;
return (unsigned int)(z - y + (z < y ? _m : 0));
}
};
// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int _m = (unsigned int)(m);
unsigned long long r = 1;
unsigned long long y = safe_mod(x, m);
while (n) {
if (n & 1) r = (r * y) % _m;
y = (y * y) % _m;
n >>= 1;
}
return r;
}
// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
if (n <= 1) return false;
if (n == 2 || n == 7 || n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = {2, 7, 61};
for (long long a : bases) {
long long t = d;
long long y = pow_mod_constexpr(a, t, n);
while (t != n - 1 && y != 1 && y != n - 1) {
y = y * y % n;
t <<= 1;
}
if (y != n - 1 && t % 2 == 0) {
return false;
}
}
return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);
// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
a = safe_mod(a, b);
if (a == 0) return {b, 0};
// Contracts:
// [1] s - m0 * a = 0 (mod b)
// [2] t - m1 * a = 0 (mod b)
// [3] s * |m1| + t * |m0| <= b
long long s = b, t = a;
long long m0 = 0, m1 = 1;
while (t) {
long long u = s / t;
s -= t * u;
m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b
// [3]:
// (s - t * u) * |m1| + t * |m0 - m1 * u|
// <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
// = s * |m1| + t * |m0| <= b
auto tmp = s;
s = t;
t = tmp;
tmp = m0;
m0 = m1;
m1 = tmp;
}
// by [3]: |m0| <= b/g
// by g != b: |m0| < b/g
if (m0 < 0) m0 += b / s;
return {s, m0};
}
// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
if (m == 2) return 1;
if (m == 167772161) return 3;
if (m == 469762049) return 3;
if (m == 754974721) return 11;
if (m == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
int x = (m - 1) / 2;
while (x % 2 == 0) x /= 2;
for (int i = 3; (long long)(i)*i <= x; i += 2) {
if (x % i == 0) {
divs[cnt++] = i;
while (x % i == 0) {
x /= i;
}
}
}
if (x > 1) {
divs[cnt++] = x;
}
for (int g = 2;; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
// @param n `n < 2^32`
// @param m `1 <= m < 2^32`
// @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)
unsigned long long floor_sum_unsigned(unsigned long long n,
unsigned long long m,
unsigned long long a,
unsigned long long b) {
unsigned long long ans = 0;
while (true) {
if (a >= m) {
ans += n * (n - 1) / 2 * (a / m);
a %= m;
}
if (b >= m) {
ans += n * (b / m);
b %= m;
}
unsigned long long y_max = a * n + b;
if (y_max < m) break;
// y_max < m * (n + 1)
// floor(y_max / m) <= n
n = (unsigned long long)(y_max / m);
b = (unsigned long long)(y_max % m);
std::swap(m, a);
}
return ans;
}
} // namespace internal
} // namespace atcoder
#line 1 "lib/ac-library/atcoder/internal_type_traits.hpp"
#line 7 "lib/ac-library/atcoder/internal_type_traits.hpp"
namespace atcoder {
namespace internal {
#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value ||
std::is_same<T, __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int128 =
typename std::conditional<std::is_same<T, __uint128_t>::value ||
std::is_same<T, unsigned __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using make_unsigned_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value,
__uint128_t,
unsigned __int128>;
template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
is_signed_int128<T>::value ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
std::is_signed<T>::value) ||
is_signed_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<(is_integral<T>::value &&
std::is_unsigned<T>::value) ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<
is_signed_int128<T>::value,
make_unsigned_int128<T>,
typename std::conditional<std::is_signed<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type>::type;
#else
template <class T> using is_integral = typename std::is_integral<T>;
template <class T>
using is_signed_int =
typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<is_integral<T>::value &&
std::is_unsigned<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type;
#endif
template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;
template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;
template <class T> using to_unsigned_t = typename to_unsigned<T>::type;
} // namespace internal
} // namespace atcoder
#line 14 "lib/ac-library/atcoder/modint.hpp"
namespace atcoder {
namespace internal {
struct modint_base {};
struct static_modint_base : modint_base {};
template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;
} // namespace internal
template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
using mint = static_modint;
public:
static constexpr int mod() { return m; }
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
static_modint() : _v(0) {}
template <class T, internal::is_signed_int_t<T>* = nullptr>
static_modint(T v) {
long long x = (long long)(v % (long long)(umod()));
if (x < 0) x += umod();
_v = (unsigned int)(x);
}
template <class T, internal::is_unsigned_int_t<T>* = nullptr>
static_modint(T v) {
_v = (unsigned int)(v % umod());
}
unsigned int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint& operator*=(const mint& rhs) {
unsigned long long z = _v;
z *= rhs._v;
_v = (unsigned int)(z % umod());
return *this;
}
mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
if (prime) {
assert(_v);
return pow(umod() - 2);
} else {
auto eg = internal::inv_gcd(_v, m);
assert(eg.first == 1);
return eg.second;
}
}
friend mint operator+(const mint& lhs, const mint& rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint& lhs, const mint& rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint& lhs, const mint& rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint& lhs, const mint& rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint& lhs, const mint& rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint& lhs, const mint& rhs) {
return lhs._v != rhs._v;
}
private:
unsigned int _v;
static constexpr unsigned int umod() { return m; }
static constexpr bool prime = internal::is_prime<m>;
};
template <int id> struct dynamic_modint : internal::modint_base {
using mint = dynamic_modint;
public:
static int mod() { return (int)(bt.umod()); }
static void set_mod(int m) {
assert(1 <= m);
bt = internal::barrett(m);
}
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
dynamic_modint() : _v(0) {}
template <class T, internal::is_signed_int_t<T>* = nullptr>
dynamic_modint(T v) {
long long x = (long long)(v % (long long)(mod()));
if (x < 0) x += mod();
_v = (unsigned int)(x);
}
template <class T, internal::is_unsigned_int_t<T>* = nullptr>
dynamic_modint(T v) {
_v = (unsigned int)(v % mod());
}
unsigned int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v += mod() - rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator*=(const mint& rhs) {
_v = bt.mul(_v, rhs._v);
return *this;
}
mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
auto eg = internal::inv_gcd(_v, mod());
assert(eg.first == 1);
return eg.second;
}
friend mint operator+(const mint& lhs, const mint& rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint& lhs, const mint& rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint& lhs, const mint& rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint& lhs, const mint& rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint& lhs, const mint& rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint& lhs, const mint& rhs) {
return lhs._v != rhs._v;
}
private:
unsigned int _v;
static internal::barrett bt;
static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt(998244353);
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;
namespace internal {
template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;
template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;
template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};
template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;
} // namespace internal
} // namespace atcoder
#line 1 "tools/ord_mod.hpp"
#include <vector>
#line 6 "tools/ord_mod.hpp"
#include <cstddef>
#include <map>
#line 1 "tools/is_prime.hpp"
#include <array>
#line 1 "tools/prod_mod.hpp"
#line 1 "tools/uint128_t.hpp"
#line 1 "tools/detail/int128_t.hpp"
#include <algorithm>
#line 7 "tools/detail/int128_t.hpp"
#include <cstdint>
#include <functional>
#line 10 "tools/detail/int128_t.hpp"
#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 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/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/prime_factorization.hpp"
#line 6 "tools/prime_factorization.hpp"
#include <queue>
#line 9 "tools/prime_factorization.hpp"
#include <cmath>
#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/pow.hpp"
#line 1 "tools/monoid.hpp"
#line 1 "tools/gcd.hpp"
#line 6 "tools/gcd.hpp"
namespace tools {
template <typename M, typename N>
constexpr ::std::common_type_t<M, N> gcd(const M m, const N n) {
return ::std::gcd(m, n);
}
}
#line 9 "tools/monoid.hpp"
namespace tools {
namespace monoid {
template <typename M, M ...dummy>
struct max;
template <typename M>
struct max<M> {
static_assert(::std::is_arithmetic_v<M>, "M must be a built-in arithmetic type.");
using T = M;
static T op(const T lhs, const T rhs) {
return ::std::max(lhs, rhs);
}
static T e() {
if constexpr (::std::is_integral_v<M>) {
return ::std::numeric_limits<M>::min();
} else {
return -::std::numeric_limits<M>::infinity();
}
}
};
template <typename M, M E>
struct max<M, E> {
static_assert(::std::is_integral_v<M>, "M must be a built-in integral type.");
using T = M;
static T op(const T lhs, const T rhs) {
assert(E <= lhs);
assert(E <= rhs);
return ::std::max(lhs, rhs);
}
static T e() {
return E;
}
};
template <typename M, M ...dummy>
struct min;
template <typename M>
struct min<M> {
static_assert(::std::is_arithmetic_v<M>, "M must be a built-in arithmetic type.");
using T = M;
static T op(const T lhs, const T rhs) {
return ::std::min(lhs, rhs);
}
static T e() {
if constexpr (::std::is_integral_v<M>) {
return ::std::numeric_limits<M>::max();
} else {
return ::std::numeric_limits<M>::infinity();
}
}
};
template <typename M, M E>
struct min<M, E> {
static_assert(::std::is_integral_v<M>, "M must be a built-in integral type.");
using T = M;
static T op(const T lhs, const T rhs) {
assert(lhs <= E);
assert(rhs <= E);
return ::std::min(lhs, rhs);
}
static T e() {
return E;
}
};
template <typename M>
struct multiplies {
private:
using VR = ::std::conditional_t<::std::is_arithmetic_v<M>, const M, const M&>;
public:
using T = M;
static T op(VR lhs, VR rhs) {
return lhs * rhs;
}
static T e() {
return T(1);
}
};
template <>
struct multiplies<bool> {
using T = bool;
static T op(const bool lhs, const bool rhs) {
return lhs && rhs;
}
static T e() {
return true;
}
};
template <typename M>
struct gcd {
private:
static_assert(!::std::is_arithmetic_v<M> || (::std::is_integral_v<M> && !::std::is_same_v<M, bool>), "If M is a built-in arithmetic type, it must be integral except for bool.");
using VR = ::std::conditional_t<::std::is_arithmetic_v<M>, const M, const M&>;
public:
using T = M;
static T op(VR lhs, VR rhs) {
return ::tools::gcd(lhs, rhs);
}
static T e() {
return T(0);
}
};
template <typename M, M E>
struct update {
static_assert(::std::is_integral_v<M>, "M must be a built-in integral type.");
using T = M;
static T op(const T lhs, const T rhs) {
return lhs == E ? rhs : lhs;
}
static T e() {
return E;
}
};
}
}
#line 1 "tools/square.hpp"
#line 1 "tools/is_monoid.hpp"
#line 6 "tools/is_monoid.hpp"
namespace tools {
template <typename M, typename = void>
struct is_monoid : ::std::false_type {};
template <typename M>
struct is_monoid<M, ::std::enable_if_t<
::std::is_same_v<typename M::T, decltype(M::op(::std::declval<typename M::T>(), ::std::declval<typename M::T>()))> &&
::std::is_same_v<typename M::T, decltype(M::e())>
, void>> : ::std::true_type {};
template <typename M>
inline constexpr bool is_monoid_v = ::tools::is_monoid<M>::value;
}
#line 6 "tools/square.hpp"
namespace tools {
template <typename M>
::std::enable_if_t<::tools::is_monoid_v<M>, typename M::T> square(const typename M::T& x) {
return M::op(x, x);
}
template <typename T>
::std::enable_if_t<!::tools::is_monoid_v<T>, T> square(const T& x) {
return x * x;
}
}
#line 9 "tools/pow.hpp"
namespace tools {
template <typename M, typename E>
::std::enable_if_t<::std::is_integral_v<E>, typename M::T> pow(const typename M::T& base, const E exponent) {
assert(exponent >= 0);
return exponent == 0
? M::e()
: exponent % 2 == 0
? ::tools::square<M>(::tools::pow<M>(base, exponent / 2))
: M::op(::tools::pow<M>(base, exponent - 1), base);
}
template <typename T, typename E>
::std::enable_if_t<::std::is_integral_v<E>, T> pow(const T& base, const E exponent) {
assert(exponent >= 0);
return ::tools::pow<::tools::monoid::multiplies<T>>(base, exponent);
}
template <typename T, typename E>
auto pow(const T base, const E exponent) -> ::std::enable_if_t<!::std::is_integral_v<E>, decltype(::std::pow(base, exponent))> {
return ::std::pow(base, exponent);
}
}
#line 1 "tools/fix.hpp"
#line 6 "tools/fix.hpp"
namespace tools {
template <typename F>
struct fix : F {
template <typename G>
fix(G&& g) : F({::std::forward<G>(g)}) {
}
template <typename... Args>
decltype(auto) operator()(Args&&... args) const {
return F::operator()(*this, ::std::forward<Args>(args)...);
}
};
template <typename F>
fix(F&&) -> fix<::std::decay_t<F>>;
}
#line 13 "tools/ord_mod.hpp"
namespace tools {
template <typename T>
class ord_mod {
private:
T m_P;
::std::vector<T> m_p;
::std::vector<T> m_e;
::std::vector<::std::vector<T>> m_E;
::std::size_t n() const {
return this->m_p.size();
}
public:
ord_mod() = default;
ord_mod(const T P) : m_P(P) {
assert(::tools::is_prime(P));
const auto factors = ::tools::prime_factorization(P - 1);
for (::std::size_t l = 0, r = 0; l < factors.size(); l = r) {
for (; r < factors.size() && factors[l] == factors[r]; ++r);
this->m_p.push_back(factors[l]);
this->m_e.push_back(r - l);
}
this->m_E.resize(this->n() + 1);
for (::std::size_t l = 0; l <= this->n(); ++l) {
this->m_E[l].resize(this->n() + 1);
this->m_E[l][l] = 1;
}
for (::std::size_t l = 0; l + 1 <= this->n(); ++l) {
this->m_E[l][l + 1] = ::tools::pow(this->m_p[l], this->m_e[l]);
}
for (::std::size_t l = 0; l + 2 <= this->n(); ++l) {
for (::std::size_t r = l + 2; r <= this->n(); ++r) {
this->m_E[l][r] = this->m_E[l][r - 1] * this->m_E[r - 1][r];
}
}
}
T query(const T x) const {
assert(1 <= x && x < this->m_P);
T m = 1;
::tools::fix([&](auto&& dfs, const ::std::size_t l, const ::std::size_t r, const T Xbar_l_r) -> void {
switch (r - l) {
case 0:
return;
case 1:
{
::std::size_t f_l;
T v;
for (f_l = 0, v = Xbar_l_r; v != 1; ++f_l, v = ::tools::pow_mod(v, this->m_p[l], this->m_P));
m *= ::tools::pow(this->m_p[l], f_l);
}
return;
default:
{
const auto m = (l + r) / 2;
dfs(l, m, ::tools::pow_mod(Xbar_l_r, this->m_E[m][r], this->m_P));
dfs(m, r, ::tools::pow_mod(Xbar_l_r, this->m_E[l][m], this->m_P));
}
return;
}
})(0, this->n(), x);
return m;
}
::std::map<T, T> count() const {
::std::vector<T> E{1};
for (const auto e_i : this->m_e) {
E.push_back(E.back() * (e_i + 1));
}
::std::vector<T> A(E.back());
A[0] = 1;
for (::std::size_t i = 0; i < this->n(); ++i) {
for (T f = 1; f <= this->m_e[i]; ++f) {
for (T s = 0; s < E[i]; ++s) {
A[f * E[i] + s] = A[(f - 1) * E[i] + s] * this->m_p[i];
}
}
}
for (auto&& A_i : A) A_i = (this->m_P - 1) / A_i;
auto B = A;
for (::std::size_t i = 0; i < this->n(); ++i) {
for (T s = 0, s_end = E.back() / E[i + 1]; s < s_end; ++s) {
for (T t = 0; t < E[i]; ++t) {
for (T f = 0; f < this->m_e[i]; ++f) {
B[s * E[i + 1] + f * E[i] + t] -= B[s * E[i + 1] + (f + 1) * E[i] + t];
}
}
}
}
::std::map<T, T> result;
for (T i = 0; i < E.back(); ++i) {
result.emplace(A[i], B[i]);
}
return result;
}
};
}
#line 7 "tests/ord_mod/count.test.cpp"
using mint = atcoder::modint998244353;
using ll = long long;
int main() {
std::cin.tie(nullptr);
std::ios_base::sync_with_stdio(false);
ll P;
std::cin >> P;
auto answer = mint::raw(1);
for (const auto& [ord, n] : tools::ord_mod(P).count()) {
answer += mint(ord) * mint(n);
}
std::cout << answer.val() << '\n';
return 0;
}