Polynomial Interpolation (Geometric Sequence) (math/fps/polynomial-interpolation-gemoetric.hpp)

• View this file on GitHub
• Last update: 2024-10-20 20:46:34+09:00
• Include: #include "math/fps/polynomial-interpolation-gemoetric.hpp"
• Link: https://noshi91.github.io/algorithm-encyclopedia/polynomial-interpolation-geometric

Depends on

• Convolution(畳み込み) (math/convolution/convolution.hpp)
• Formal Power Series(形式的冪級数) (math/fps/fps.hpp)
• Multipoint Evaluation (Geometric Sequence) (math/fps/multipoint-evaluation-geometric.hpp)
• ModInt (math/modular/modint.hpp)
• MontgomeryModInt(モンゴメリ乗算) (math/modular/montgomery-modint.hpp)
• Miller-Rabin Primality Test(ミラーラビン素数判定) (math/number/miller-rabin.hpp)
• Pollard's Rho Factorization(ポラード・ロー法) (math/number/pollard-rho.hpp)
• Primitive Root(原始根) (math/number/primitive-root.hpp)
• Combinatorics(組み合わせ) (math/others/combinatorics.hpp)
• Random(乱数) (others/random.hpp)
• string/run-length.hpp
• template/alias.hpp
• template/func.hpp
• template/in.hpp
• template/macro.hpp
• template/out.hpp
• template/template.hpp
• template/type-traits.hpp
• template/util.hpp

Required by

• scrap/static-top-tree.hpp

Verified with

• test/yosupo/polynomial/polynomial_interpolation_on_geometric_sequence.test.cpp

Code

#pragma once
#include "fps.hpp"
#include "multipoint-evaluation-geometric.hpp"

template <typename T>
FormalPowerSeries<T> polynomial_interpolation_gemoetric(vector<T> y, const T& a, const T& r) {
    const int n = y.size();
    if (n == 0) return {};
    if (n == 1) return {y[0]};
    if (r == 0) {
        return {y[1], (y[0] - y[1]) / a};
    }
    vector<T> pow_r(n + 1), pow_invr(n + 1), inv_s(n);
    T s = 1;
    const T inv_r = T(1) / r;
    {
        pow_r[0] = pow_invr[0] = inv_s[0] = 1;
        rep(i, n) {
            if (i) s *= (1 - pow_r[i]);
            pow_r[i + 1] = pow_r[i] * r;
            pow_invr[i + 1] = pow_invr[i] * inv_r;
        }
        inv_s[n - 1] = T(1) / s;
        s *= (1 - pow_r[n]);
        rrep(i, n - 1) inv_s[i] = inv_s[i + 1] * (1 - pow_r[i + 1]);
    }
    T v = 1;
    rep(i, n) {
        y[i] *= v * inv_s[i] * inv_s[n - i - 1];
        if (i & 1) y[i] = -y[i];
        if (i != n - 1) v *= pow_invr[n - 2 - i];
    }
    FormalPowerSeries<T> f1(n + 1);
    v = 1;
    rep(i, n + 1) {
        if (i == 0 || i == n)
            f1[i] = v;
        else
            f1[i] = v * s * inv_s[i] * inv_s[n - i];
        if (i & 1) f1[i] = -f1[i];
        v *= pow_r[i];
    }
    FormalPowerSeries<T> f2 = multipoint_evaluation_geometric(FormalPowerSeries<T>(y), T(1), r, n);
    auto res = (f1 * f2);
    res.resize(n);
    reverse(res.begin(), res.end());
    const T inva = T(1) / a;
    v = 1;
    rep(i, n) res[i] *= v, v *= inva;
    return res;
}
/**
 * @brief Polynomial Interpolation (Geometric Sequence)
 * @ref https://noshi91.github.io/algorithm-encyclopedia/polynomial-interpolation-geometric
 */

#line 2 "template/template.hpp"
#include <bits/stdc++.h>

#line 3 "template/alias.hpp"

using ll = long long;
using ull = unsigned long long;
using ld = long double;
using i128 = __int128_t;
using u128 = __uint128_t;
using pi = std::pair<int, int>;
using pl = std::pair<ll, ll>;
using vi = std::vector<int>;
using vl = std::vector<ll>;
using vs = std::vector<std::string>;
using vc = std::vector<char>;
using vvl = std::vector<vl>;
using vd = std::vector<double>;
using vp = std::vector<pl>;
using vb = std::vector<bool>;
template <typename T>
struct infinity {
    static constexpr T max = std::numeric_limits<T>::max();
    static constexpr T min = std::numeric_limits<T>::min();
    static constexpr T value = std::numeric_limits<T>::max() / 2;
    static constexpr T mvalue = std::numeric_limits<T>::min() / 2;
};
template <typename T>
constexpr T INF = infinity<T>::value;
constexpr ll inf = INF<ll>;
constexpr ld EPS = 1e-8;
constexpr ld PI = 3.1415926535897932384626;
constexpr int dx[8] = {-1, 0, 1, 0, 1, -1, -1, 1};
constexpr int dy[8] = {0, 1, 0, -1, 1, 1, -1, -1};
#line 3 "template/macro.hpp"

#ifndef __COUNTER__
#define __COUNTER__ __LINE__
#endif

#define SELECT4(a, b, c, d, e, ...) e
#define SELECT3(a, b, c, d, ...) d
#define REP_1(a, c) for (ll REP_##c = 0; REP_##c < (ll)(a); ++REP_##c)
#define REP1(a) REP_1(a, __COUNTER__)
#define REP2(i, a) for (ll i = 0; i < (ll)(a); ++i)
#define REP3(i, a, b) for (ll i = (ll)(a); i < (ll)(b); ++i)
#define REP4(i, a, b, c) for (ll i = (ll)(a); i < (ll)(b); i += (ll)(c))
#define rep(...) SELECT4(__VA_ARGS__, REP4, REP3, REP2, REP1)(__VA_ARGS__)
#define RREP_1(a, c) for (ll RREP_##c = (ll)(a) - 1; RREP_##c >= 0; --RREP_##c)
#define RREP1(a) RREP_1(a, __COUNTER__)
#define RREP2(i, a) for (ll i = (ll)(a) - 1; i >= 0; --i)
#define RREP3(i, a, b) for (ll i = (ll)(b) - 1; i >= (ll)(a); --i)
#define rrep(...) SELECT3(__VA_ARGS__, RREP3, RREP2, RREP1)(__VA_ARGS__)
#define all(v) std::begin(v), std::end(v)
#define rall(v) std::rbegin(v), std::rend(v)
#define INT(...)     \
    int __VA_ARGS__; \
    scan(__VA_ARGS__)
#define LL(...)     \
    ll __VA_ARGS__; \
    scan(__VA_ARGS__)
#define STR(...)        \
    string __VA_ARGS__; \
    scan(__VA_ARGS__)
#define CHR(...)      \
    char __VA_ARGS__; \
    scan(__VA_ARGS__)
#define DBL(...)        \
    double __VA_ARGS__; \
    scan(__VA_ARGS__)
#define LD(...)     \
    ld __VA_ARGS__; \
    scan(__VA_ARGS__)
#define pb push_back
#define eb emplace_back
#line 3 "template/type-traits.hpp"

#line 5 "template/type-traits.hpp"

template <typename T, typename... Args>
struct function_traits_impl {
    using return_type = T;
    static constexpr std::size_t arg_size = sizeof...(Args);
    template <std::size_t idx>
    using argument_type = typename std::tuple_element<idx, std::tuple<Args...>>::type;
    using argument_types = std::tuple<Args...>;
};

template <typename>
struct function_traits_helper;
template <typename T, typename Tp, typename... Args>
struct function_traits_helper<T (Tp::*)(Args...)> : function_traits_impl<T, Args...> {};
template <typename T, typename Tp, typename... Args>
struct function_traits_helper<T (Tp::*)(Args...) const> : function_traits_impl<T, Args...> {};
template <typename T, typename Tp, typename... Args>
struct function_traits_helper<T (Tp::*)(Args...)&> : function_traits_impl<T, Args...> {};
template <typename T, typename Tp, typename... Args>
struct function_traits_helper<T (Tp::*)(Args...) const&> : function_traits_impl<T, Args...> {};

template <typename F>
using function_traits = function_traits_helper<decltype(&std::remove_reference<F>::type::operator())>;
template <typename F>
using function_return_type = typename function_traits<F>::return_type;
template <typename F, std::size_t idx>
using function_argument_type = typename function_traits<F>::template argument_type<idx>;
template <typename F>
using function_argument_types = typename function_traits<F>::argument_types;
template <class T>
using is_signed_int = std::integral_constant<bool, (std::is_integral<T>::value && std::is_signed<T>::value) || std::is_same<T, __int128_t>::value>;
template <class T>
using is_unsigned_int = std::integral_constant<bool, (std::is_integral<T>::value && std::is_unsigned<T>::value) || std::is_same<T, __uint128_t>::value>;
template <class T>
using is_int = std::integral_constant<bool, is_signed_int<T>::value || is_unsigned_int<T>::value>;
template <typename T, typename = void>
struct is_range : std::false_type {};
template <typename T>
struct is_range<
    T,
    decltype(all(std::declval<typename std::add_lvalue_reference<T>::type>()), (void)0)> : std::true_type {};
template <std::size_t size>
struct int_least {
    static_assert(size <= 128, "size must be less than or equal to 128");
    using type = typename std::conditional<
        size <= 8, std::int_least8_t,
        typename std::conditional<
            size <= 16, std::int_least16_t,
            typename std::conditional<
                size <= 32, std::int_least32_t,
                typename std::conditional<size <= 64, std::int_least64_t, __int128_t>::type>::type>::type>::type;
};
template <std::size_t size>
using int_least_t = typename int_least<size>::type;
template <std::size_t size>
struct uint_least {
    static_assert(size <= 128, "size must be less than or equal to 128");
    using type = typename std::conditional<
        size <= 8, std::uint_least8_t,
        typename std::conditional<
            size <= 16, std::uint_least16_t,
            typename std::conditional<
                size <= 32, std::uint_least32_t,
                typename std::conditional<size <= 64, std::uint_least64_t, __uint128_t>::type>::type>::type>::type;
};
template <std::size_t size>
using uint_least_t = typename uint_least<size>::type;
template <typename T>
using double_size_int = int_least<std::numeric_limits<T>::digits * 2 + 1>;
template <typename T>
using double_size_int_t = typename double_size_int<T>::type;
template <typename T>
using double_size_uint = uint_least<std::numeric_limits<T>::digits * 2>;
template <typename T>
using double_size_uint_t = typename double_size_uint<T>::type;
template <typename T>
using double_size = typename std::conditional<std::is_signed<T>::value, double_size_int<T>, double_size_uint<T>>::type;
template <typename T>
using double_size_t = typename double_size<T>::type;
#line 2 "template/in.hpp"
#include <unistd.h>
#line 5 "template/in.hpp"
namespace fastio {
template <std::size_t BUFF_SIZE = 1 << 17, int decimal_precision = 16>
struct Scanner {
   private:
    template <typename, typename = void>
    struct has_scan : std::false_type {};
    template <class T>
    struct has_scan<T, decltype(std::declval<T>().scan(std::declval<Scanner&>()), (void)0)> : std::true_type {};
    int fd;
    char buffer[BUFF_SIZE + 1];
    int idx, sz;
    bool state;
    inline void load() {
        int len = sz - idx;
        if (idx < len) return;
        std::memcpy(buffer, buffer + idx, len);
        sz = len + read(fd, buffer + len, BUFF_SIZE - len);
        idx = 0;
        buffer[sz] = 0;
    }
    inline char cur() {
        if (idx == sz) load();
        if (idx == sz) {
            state = false;
            return '\0';
        }
        return buffer[idx];
    }
    inline void next() {
        if (idx == sz) load();
        if (idx == sz) return;
        idx++;
    }

   public:
    Scanner() : Scanner(0) {}
    explicit Scanner(int fd) : fd(fd), idx(0), sz(0), state(true) {}
    explicit Scanner(FILE* file) : fd(fileno(file)), idx(0), sz(0), state(true) {}

    inline char scan_char() {
        if (idx == sz) load();
        return (idx == sz ? '\0' : buffer[idx++]);
    }
    Scanner ignore(int n = 1) {
        if (idx + n > sz) load();
        idx += n;
        return (*this);
    }
    inline void skip_space() {
       
        if (idx == sz) load();
        while (('\t' <= cur() && cur() <= '\r') || cur() == ' ') {
            if (++idx == sz) load();
        }
    }
    void scan(char& a) {
        skip_space();
        a = scan_char();
    }
    void scan(std::string& a) {
        skip_space();
        a.clear();
        while (cur() != '\0' && (buffer[idx] < '\t' || '\r' < buffer[idx]) && buffer[idx] != ' ') {
            a += scan_char();
        }
    }
    template <std::size_t len>
    void scan(std::bitset<len>& a) {
        skip_space();
        if (idx + len > sz) load();
        rrep(i, len) a[i] = (buffer[idx++] != '0');
    }
    template <typename T, typename std::enable_if<is_int<T>::value && !has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        skip_space();
        bool neg = false;
        if constexpr (std::is_signed<T>::value || std::is_same_v<T, __int128_t>) {
            if (cur() == '-') {
                neg = true;
                next();
            }
        }
        if (idx + 40 > sz && (idx == sz || ('0' <= buffer[sz - 1] && buffer[sz - 1] <= '9'))) load();
        a = 0;
        while ('0' <= buffer[idx] && buffer[idx] <= '9') {
            a = a * 10 + (buffer[idx++] & 15);
        }
        if constexpr (std::is_signed<T>::value || std::is_same<T, __int128_t>::value) {
            if (neg) a = -a;
        }
    }
    template <typename T, typename std::enable_if<std::is_floating_point<T>::value && !has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        skip_space();
        bool neg = false;
        if (cur() == '-') {
            neg = true;
            next();
        }
        a = 0;
        while ('0' <= cur() && cur() <= '9') {
            a = a * 10 + (scan_char() & 15);
        }
        if (cur() == '.') {
            next();
            T n = 0, d = 1;
            for (int i = 0; '0' <= cur() && cur() <= '9' && i < decimal_precision; ++i) {
                n = n * 10 + (scan_char() & 15);
                d *= 10;
            }
            while ('0' <= cur() && cur() <= '9') next();
            a += n / d;
        }
        if (neg) a = -a;
    }

   private:
    template <std::size_t i, typename... Args>
    void scan(std::tuple<Args...>& a) {
        if constexpr (i < sizeof...(Args)) {
            scan(std::get<i>(a));
            scan<i + 1, Args...>(a);
        }
    }

   public:
    template <typename... Args>
    void scan(std::tuple<Args...>& a) {
        scan<0, Args...>(a);
    }
    template <typename T, typename U>
    void scan(std::pair<T, U>& a) {
        scan(a.first);
        scan(a.second);
    }
    template <typename T, typename std::enable_if<is_range<T>::value && !has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        for (auto& i : a) scan(i);
    }
    template <typename T, typename std::enable_if<has_scan<T>::value>::type* = nullptr>
    void scan(T& a) {
        a.scan(*this);
    }
    void operator()() {}
    template <typename Head, typename... Tail>
    void operator()(Head& head, Tail&... tail) {
        scan(head);
        operator()(std::forward<Tail&>(tail)...);
    }
    template <typename T>
    Scanner& operator>>(T& a) {
        scan(a);
        return *this;
    }
    explicit operator bool() const { return state; }
    friend Scanner& getline(Scanner& sc, std::string& a) {
        a.clear();
        char c;
        if ((c = sc.scan_char()) == '\0' || c == '\n') return sc;
        a += c;
        while ((c = sc.scan_char()) != '\0' && c != '\n') a += c;
        return sc;
    }
};
Scanner<> sc;
}  // namespace fastio
using fastio::sc;
#line 6 "template/out.hpp"

namespace fastio {
struct Pre {
    char buffer[10000][4];
    constexpr Pre() : buffer() {
        for (int i = 0; i < 10000; ++i) {
            int n = i;
            for (int j = 3; j >= 0; --j) {
                buffer[i][j] = n % 10 | '0';
                n /= 10;
            }
        }
    }
} constexpr pre;
template <std::size_t BUFF_SIZE = 1 << 17, bool debug = false>
struct Printer {
   private:
    template <typename, bool = debug, class = void>
    struct has_print : std::false_type {};
    template <typename T>
    struct has_print<T, false, decltype(std::declval<T>().print(std::declval<Printer&>()), (void)0)> : std::true_type {};
    template <typename T>
    struct has_print<T, true, decltype(std::declval<T>().debug(std::declval<Printer&>()), (void)0)> : std::true_type {};
    int fd;
    char buffer[BUFF_SIZE];
    int idx;
    std::size_t decimal_precision;

   public:
    Printer() : Printer((debug ? 2 : 1)) {}
    explicit Printer(int fd) : fd(fd), idx(0), decimal_precision(16) {}
    explicit Printer(FILE* file) : fd(fileno(file)), idx(0), decimal_precision(16) {}
    ~Printer() {
        flush();
    }
    void set_decimal_precision(std::size_t n) { decimal_precision = n; }
    inline void print_char(char c) {
        buffer[idx++] = c;
        if (idx == BUFF_SIZE) flush();
    }
    inline void flush() {
        idx = write(fd, buffer, idx);
        idx = 0;
    }
    void print(char a) {
        if constexpr (debug) print_char('\'');
        print_char(a);
        if constexpr (debug) print_char('\'');
    }
    void print(bool a) {
        if constexpr (debug) print_char('\'');
        print_char('0' + a);
        if constexpr (debug) print_char('\'');
    }
    void print(const char* a) {
        if constexpr (debug) print_char('\"');
        for (; *a != '\0'; ++a) print_char(*a);
        if constexpr (debug) print_char('\"');
    }
    template <std::size_t N>
    void print(const char (&a)[N]) {
        if constexpr (debug) print_char('\"');
        for (auto i : a) print_char(i);
        if constexpr (debug) print_char('\"');
    }
    void print(const std::string& a) {
        if constexpr (debug) print_char('\"');
        for (auto i : a) print_char(i);
        if constexpr (debug) print_char('\"');
    }
    template <std::size_t len>
    void print(const std::bitset<len>& a) {
        for (int i = len - 1; i >= 0; --i) print_char('0' + a[i]);
    }
    template <typename T, typename std::enable_if<is_int<T>::value && !has_print<T>::value>::type* = nullptr>
    void print(T a) {
        if (!a) {
            print_char('0');
            return;
        }
        if constexpr (is_signed_int<T>::value) {
            if (a < 0) {
                print_char('-');
                a = -a;
            }
        }
        if (static_cast<size_t>(idx + 40) >= BUFF_SIZE) flush();
        static char stk[40];
        int top = 40;
        while (a >= 10000) {
            int i = a % 10000;
            a /= 10000;
            top -= 4;
            std::memcpy(stk + top, pre.buffer[i], 4);
        }
        if (a >= 1000) {
            std::memcpy(buffer + idx, pre.buffer[a], 4);
            idx += 4;
        } else if (a >= 100) {
            std::memcpy(buffer + idx, pre.buffer[a] + 1, 3);
            idx += 3;
        } else if (a >= 10) {
            std::memcpy(buffer + idx, pre.buffer[a] + 2, 2);
            idx += 2;
        } else {
            buffer[idx++] = '0' | a;
        }
        std::memcpy(buffer + idx, stk + top, 40 - top);
        idx += 40 - top;
    }
    template <typename T, typename std::enable_if<std::is_floating_point<T>::value && !has_print<T>::value>::type* = nullptr>
    void print(T a) {
        if (a == infinity<T>::max || a == infinity<T>::value) {
            print("inf");
            return;
        }
        if (a == infinity<T>::min || a == infinity<T>::mvalue) {
            print("-inf");
            return;
        }
        if (std::isnan(a)) {
            print("nan");
            return;
        }
        if (a < 0) {
            print_char('-');
            a = -a;
        }
        T b = a;
        if (b < 1) {
            print_char('0');
        } else {
            std::string s;
            while (b >= 1) {
                s += (char)('0' | (int)std::fmod(b, 10.0));
                b /= 10;
            }
            for (auto i = s.rbegin(); i != s.rend(); ++i) {
                print_char(*i);
            }
        }
        print_char('.');
        for (std::size_t _ = 0; _ < decimal_precision; ++_) {
            a *= 10;
            print_char('0' | (int)std::fmod(a, 10.0));
        }
    }

   private:
    template <std::size_t i, typename... Args>
    void print(const std::tuple<Args...>& a) {
        if constexpr (i < sizeof...(Args)) {
            if constexpr (debug) print_char(',');
            print_char(' ');
            print(std::get<i>(a));
            print<i + 1>(a);
        }
    }

   public:
    template <typename... Args>
    void print(const std::tuple<Args...>& a) {
        if constexpr (debug) print_char('(');
        if constexpr (sizeof...(Args) != 0) {
            print(std::get<0>(a));
        }
        print<1, Args...>(a);
        if constexpr (debug) print_char(')');
    }
    template <typename T, typename U>
    void print(const std::pair<T, U>& a) {
        if constexpr (debug) print_char('(');
        print(a.first);
        if constexpr (debug) print_char(',');
        print_char(' ');
        print(a.second);
        if constexpr (debug) print_char(')');
    }
    template <typename T, typename std::enable_if<is_range<T>::value>::type* = nullptr>
    void print(const T& a) {
        if constexpr (debug) print_char('{');
        auto it = std::begin(a);
        if (it != std::end(a)) {
            print(*it);
            while (++it != std::end(a)) {
                if constexpr (debug) print_char(',');
                print_char(' ');
                print(*it);
            }
        }
        if constexpr (debug) print_char('}');
    }
    template <typename T, typename std::enable_if<has_print<T>::value && !debug>::type* = nullptr>
    void print(const T& a) {
        a.print(*this);
    }
    template <typename T, typename std::enable_if<has_print<T>::value && debug>::type* = nullptr>
    void print(const T& a) {
        a.debug(*this);
    }
    void operator()() {}
    template <typename Head, typename... Tail>
    void operator()(const Head& head, const Tail&... tail) {
        print(head);
        operator()(std::forward<const Tail&>(tail)...);
    }
    template <typename T>
    Printer& operator<<(const T& a) {
        print(a);
        return *this;
    }
    Printer& operator<<(Printer& (*f)(Printer&)) {
        return f(*this);
    }
};

template <std::size_t BUFF_SIZE, bool debug>
Printer<BUFF_SIZE, debug>& endl(Printer<BUFF_SIZE, debug>& out) {
    out.print_char('\n');
    out.flush();
    return out;
}
template <std::size_t BUFF_SIZE, bool debug>
Printer<BUFF_SIZE, debug>& flush(Printer<BUFF_SIZE, debug>& out) {
    out.flush();
    return out;
}
Printer<> pr;
Printer<1 << 17, true> prd;
}  // namespace fastio
using fastio::endl;
using fastio::flush;
using fastio::pr;
using fastio::prd;
#line 3 "template/func.hpp"

#line 8 "template/func.hpp"

inline constexpr int msb(ull x) {
    int res = x ? 0 : -1;
    if (x & 0xffffffff00000000) x &= 0xffffffff00000000, res += 32;
    if (x & 0xffff0000ffff0000) x &= 0xffff0000ffff0000, res += 16;
    if (x & 0xff00ff00ff00ff00) x &= 0xff00ff00ff00ff00, res += 8;
    if (x & 0xf0f0f0f0f0f0f0f0) x &= 0xf0f0f0f0f0f0f0f0, res += 4;
    if (x & 0xcccccccccccccccc) x &= 0xcccccccccccccccc, res += 2;
    return res + (x & 0xaaaaaaaaaaaaaaaa ? 1 : 0);
}
inline constexpr int ceil_log2(ull x) { return x ? msb(x - 1) + 1 : 0; }
inline constexpr ull reverse(ull x) {
    x = ((x & 0x5555555555555555) << 1) | ((x & 0xaaaaaaaaaaaaaaaa) >> 1);
    x = ((x & 0x3333333333333333) << 2) | ((x & 0xcccccccccccccccc) >> 2);
    x = ((x & 0x0f0f0f0f0f0f0f0f) << 4) | ((x & 0xf0f0f0f0f0f0f0f0) >> 4);
    x = ((x & 0x00ff00ff00ff00ff) << 8) | ((x & 0xff00ff00ff00ff00) >> 8);

    x = ((x & 0x0000ffff0000ffff) << 16) | ((x & 0xffff0000ffff0000) >> 16);
    return (x << 32) | (x >> 32);
}
inline constexpr ull reverse(ull x, int len) { return reverse(x) >> (64 - len); }
inline constexpr int popcnt(ull x) {
#if __cplusplus >= 202002L
    return std::popcount(x);
#endif
    x = (x & 0x5555555555555555) + ((x >> 1) & 0x5555555555555555);
    x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333);
    x = (x & 0x0f0f0f0f0f0f0f0f) + ((x >> 4) & 0x0f0f0f0f0f0f0f0f);
    x = (x & 0x00ff00ff00ff00ff) + ((x >> 8) & 0x00ff00ff00ff00ff);
    x = (x & 0x0000ffff0000ffff) + ((x >> 16) & 0x0000ffff0000ffff);
    return (x & 0x00000000ffffffff) + ((x >> 32) & 0x00000000ffffffff);
}
template <typename T, typename U>
inline constexpr bool chmin(T& a, U b) { return a > b && (a = b, true); }
template <typename T, typename U>
inline constexpr bool chmax(T& a, U b) { return a < b && (a = b, true); }
inline constexpr ll gcd(ll a, ll b) {
    if (a < 0) a = -a;
    if (b < 0) b = -b;
    while (b) {
        const ll c = b;
        b = a % b;
        a = c;
    }
    return a;
}
inline constexpr ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
inline constexpr bool is_prime(ll n) {
    if (n <= 1) return false;
    for (ll i = 2; i * i <= n; i++) {
        if (n % i == 0) return false;
    }
    return true;
}
inline constexpr ll my_pow(ll a, ll b) {
    ll res = 1;
    while (b) {
        if (b & 1) res *= a;
        a *= a;
        b >>= 1;
    }
    return res;
}
inline constexpr ll mod_pow(ll a, ll b, const ll& mod) {
    if (mod == 1) return 0;
    a %= mod;
    ll res = 1;
    while (b) {
        if (b & 1) (res *= a) %= mod;
        (a *= a) %= mod;
        b >>= 1;
    }
    return res;
}
inline ll mod_inv(ll a, const ll& mod) {
    ll b = mod, x = 1, u = 0, t;
    while (b) {
        t = a / b;
        std::swap(a -= t * b, b);
        std::swap(x -= t * u, u);
    }
    if (x < 0) x += mod;
    return x;
}
template <typename T, typename U>
std::ostream& operator<<(std::ostream& os, const std::pair<T, U>& p) {
    os << p.first << " " << p.second;
    return os;
}
template <typename T, typename U>
std::istream& operator>>(std::istream& is, std::pair<T, U>& p) {
    is >> p.first >> p.second;
    return is;
}
template <typename T>
std::ostream& operator<<(std::ostream& os, const std::vector<T>& v) {
    for (auto it = std::begin(v); it != std::end(v);) {
        os << *it << ((++it) != std::end(v) ? " " : "");
    }
    return os;
}
template <typename T>
std::istream& operator>>(std::istream& is, std::vector<T>& v) {
    for (T& in : v) {
        is >> in;
    }
    return is;
}
inline void scan() {}
template <class Head, class... Tail>
inline void scan(Head& head, Tail&... tail) {
    sc >> head;
    scan(tail...);
}
template <class T>
inline void print(const T& t) { pr << t << '\n'; }
template <class Head, class... Tail>
inline void print(const Head& head, const Tail&... tail) {
    pr << head << ' ';
    print(tail...);
}
template <class... T>
inline void fin(const T&... a) {
    print(a...);
    exit(0);
}
template <typename T>
inline void dump(const T& a) { prd << a; }
inline void trace() { prd << endl; }
template <typename Head, typename... Tail>
inline void trace(const Head& head, const Tail&... tail) {
    dump(head);
    if (sizeof...(tail)) prd.print_char(','), prd.print_char(' ');
    trace(tail...);
}
#ifdef ONLINE_JUDGE
#define dbg(...) (void(0))
#else
#define dbg(...)                                                       \
    do {                                                               \
        prd << #__VA_ARGS__;                                           \
        prd.print_char(' '), prd.print_char('='), prd.print_char(' '); \
        trace(__VA_ARGS__);                                            \
    } while (0)
#endif
#line 3 "template/util.hpp"

#line 6 "template/util.hpp"
template <typename F>
struct REC {
   private:
    F f;

   public:
    explicit constexpr REC(F&& f_) : f(std::forward<F>(f_)) {}
    template <typename... Args>
    constexpr auto operator()(Args&&... args) const {
        return f(*this, std::forward<Args>(args)...);
    }
};
template <typename T, typename Comp = std::less<T>>
struct compressor {
   private:
    std::vector<T> data;
    Comp cmp;
    bool sorted = false;

   public:
    compressor() : compressor(Comp()) {}
    compressor(const Comp& cmp) : cmp(cmp) {}
    compressor(const std::vector<T>& dat, const Comp& cmp = Comp()) : data(dat), cmp(cmp) {}
    compressor(std::vector<T>&& dat, const Comp& cmp = Comp()) : data(std::move(dat)), cmp(cmp) {}
    compressor(std::initializer_list<T> li, const Comp& cmp = Comp()) : data(li.begin(), li.end()), cmp(cmp) {}
    void push_back(const T& v) {
        assert(!sorted);
        data.push_back(v);
    }
    void push_back(T&& v) {
        assert(!sorted);
        data.push_back(std::move(v));
    }
    template <typename... Args>
    void emplace_back(Args&&... args) {
        assert(!sorted);
        data.emplace_back(std::forward<Args>(args)...);
    }
    void push(const std::vector<T>& v) {
        assert(!sorted);
        const int n = data.size();
        data.resize(v.size() + n);
        for (int i = 0; i < (int)v.size(); i++) data[i + n] = v[i];
    }
    void build() {
        assert(!sorted);
        sorted = 1;
        std::sort(data.begin(), data.end(), cmp);
        data.erase(unique(data.begin(), data.end(), [&](const T& l, const T& r) -> bool { return !cmp(l, r) && !cmp(r, l); }), data.end());
    }
    const T& operator[](int k) const& {
        assert(sorted);
        return data[k];
    }
    int get_index(const T& v) const {
        assert(sorted);
        return int(lower_bound(data.begin(), data.end(), v, cmp) - data.begin());
    }
    void press(std::vector<T>& v) const {
        assert(sorted);
        for (auto&& i : v) i = get_index(i);
    }
    std::vector<int> pressed(const std::vector<T>& v) const {
        assert(sorted);
        std::vector<int> ret(v.size());
        for (int i = 0; i < (int)v.size(); i++) ret[i] = get_index(v[i]);
        return ret;
    }
    int size() const {
        assert(sorted);
        return data.size();
    }
};
#line 11 "template/template.hpp"
using namespace std;
#line 3 "math/modular/modint.hpp"

namespace internal {
struct modint_base {};
}  // namespace internal
template <typename T>
using is_modint = is_base_of<internal::modint_base, T>;
template <typename T, T mod>
struct StaticModInt : internal::modint_base {
    static_assert(is_integral<T>::value, "T must be integral");
    static_assert(is_unsigned<T>::value, "T must be unsgined");
    static_assert(mod > 0, "mod must be positive");
    static_assert(mod <= INF<T>, "mod*2 must be less than or equal to T::max()");

   private:
    using large_t = typename double_size_uint<T>::type;
    using signed_t = typename make_signed<T>::type;
    T val;

   public:
    constexpr StaticModInt() : val(0) {}
    template <typename U, typename enable_if<is_integral<U>::value && is_unsigned<U>::value>::type* = nullptr>
    constexpr StaticModInt(U x) : val(x % mod) {}
    template <typename U, typename enable_if<is_integral<U>::value && is_signed<U>::value>::type* = nullptr>
    constexpr StaticModInt(U x) : val{} {
        x %= static_cast<signed_t>(mod);
        if (x < 0) x += static_cast<signed_t>(mod);
        val = static_cast<T>(x);
    }
    constexpr T get() const { return val; }
    static constexpr T get_mod() { return mod; }
    static constexpr StaticModInt raw(T v) {
        StaticModInt res;
        res.val = v;
        return res;
    }
    constexpr StaticModInt inv() const {
        return mod_inv(val, mod);
    }
    constexpr StaticModInt& operator++() {
        ++val;
        if (val == mod) val = 0;
        return *this;
    }
    constexpr StaticModInt operator++(int) {
        StaticModInt res = *this;
        ++*this;
        return res;
    }
    constexpr StaticModInt& operator--() {
        if (val == 0) val = mod;
        --val;
        return *this;
    }
    constexpr StaticModInt operator--(int) {
        StaticModInt res = *this;
        --*this;
        return res;
    }
    constexpr StaticModInt& operator+=(const StaticModInt& x) {
        val += x.val;
        if (val >= mod) val -= mod;
        return *this;
    }
    constexpr StaticModInt& operator-=(const StaticModInt& x) {
        if (val < x.val) val += mod;
        val -= x.val;
        return *this;
    }
    constexpr StaticModInt& operator*=(const StaticModInt& x) {
        val = static_cast<T>((static_cast<large_t>(val) * x.val) % mod);
        return *this;
    }
    constexpr StaticModInt& operator/=(const StaticModInt& x) {
        return *this *= x.inv();
    }
    friend constexpr StaticModInt operator+(const StaticModInt& l, const StaticModInt& r) { return StaticModInt(l) += r; }
    friend constexpr StaticModInt operator-(const StaticModInt& l, const StaticModInt& r) { return StaticModInt(l) -= r; }
    friend constexpr StaticModInt operator*(const StaticModInt& l, const StaticModInt& r) { return StaticModInt(l) *= r; }
    friend constexpr StaticModInt operator/(const StaticModInt& l, const StaticModInt& r) { return StaticModInt(l) /= r; }
    constexpr StaticModInt operator+() const { return StaticModInt(*this); }
    constexpr StaticModInt operator-() const { return StaticModInt() - *this; }
    friend constexpr bool operator==(const StaticModInt& l, const StaticModInt& r) { return l.val == r.val; }
    friend constexpr bool operator!=(const StaticModInt& l, const StaticModInt& r) { return l.val != r.val; }
    constexpr StaticModInt pow(ll a) const {
        StaticModInt v = *this, res = 1;
        while (a) {
            if (a & 1) res *= v;
            v *= v;
            a >>= 1;
        }
        return res;
    }
    template <typename Sc>
    void scan(Sc& a) {
        ll x;
        a.scan(x);
        *this = x;
    }
    template <typename Pr>
    void print(Pr& a) const {
        a.print(val);
    }
    template <typename Pr>
    void debug(Pr& a) const {
        a.print(val);
    }
};
template <unsigned int p>
using ModInt = StaticModInt<unsigned int, p>;

template <typename T, int id>
struct DynamicModInt {
    static_assert(is_integral<T>::value, "T must be integral");
    static_assert(is_unsigned<T>::value, "T must be unsigned");

   private:
    using large_t = typename double_size_uint<T>::type;
    using signed_t = typename make_signed<T>::type;
    T val;
    static T mod;

   public:
    constexpr DynamicModInt() : val(0) {}
    template <typename U, typename enable_if<is_integral<U>::value && is_unsigned<U>::value>::type* = nullptr>
    constexpr DynamicModInt(U x) : val(x % mod) {}
    template <typename U, typename enable_if<is_integral<U>::value && is_signed<U>::value>::type* = nullptr>
    constexpr DynamicModInt(U x) : val{} {
        x %= static_cast<signed_t>(mod);
        if (x < 0) x += static_cast<signed_t>(mod);
        val = static_cast<T>(x);
    }
    T get() const { return val; }
    static T get_mod() { return mod; }
    static void set_mod(T x) {
        mod = x;
        assert(mod > 0);
        assert(mod <= INF<T>);
    }
    static DynamicModInt raw(T v) {
        DynamicModInt res;
        res.val = v;
        return res;
    }
    DynamicModInt inv() const {
        return mod_inv(val, mod);
    }
    DynamicModInt& operator++() {
        ++val;
        if (val == mod) val = 0;
        return *this;
    }
    DynamicModInt operator++(int) {
        DynamicModInt res = *this;
        ++*this;
        return res;
    }
    DynamicModInt& operator--() {
        if (val == 0) val = mod;
        --val;
        return *this;
    }
    DynamicModInt operator--(int) {
        DynamicModInt res = *this;
        --*this;
        return res;
    }
    DynamicModInt& operator+=(const DynamicModInt& x) {
        val += x.val;
        if (val >= mod) val -= mod;
        return *this;
    }
    DynamicModInt& operator-=(const DynamicModInt& x) {
        if (val < x.val) val += mod;
        val -= x.val;
        return *this;
    }
    DynamicModInt& operator*=(const DynamicModInt& x) {
        val = static_cast<T>((static_cast<large_t>(val) * x.val) % mod);
        return *this;
    }
    DynamicModInt& operator/=(const DynamicModInt& x) {
        return *this *= x.inv();
    }
    friend DynamicModInt operator+(const DynamicModInt& l, const DynamicModInt& r) { return DynamicModInt(l) += r; }
    friend DynamicModInt operator-(const DynamicModInt& l, const DynamicModInt& r) { return DynamicModInt(l) -= r; }
    friend DynamicModInt operator*(const DynamicModInt& l, const DynamicModInt& r) { return DynamicModInt(l) *= r; }
    friend DynamicModInt operator/(const DynamicModInt& l, const DynamicModInt& r) { return DynamicModInt(l) /= r; }
    DynamicModInt operator+() const { return DynamicModInt(*this); }
    DynamicModInt operator-() const { return DynamicModInt() - *this; }
    friend bool operator==(const DynamicModInt& l, const DynamicModInt& r) { return l.val == r.val; }
    friend bool operator!=(const DynamicModInt& l, const DynamicModInt& r) { return l.val != r.val; }
    DynamicModInt pow(ll a) const {
        DynamicModInt v = *this, res = 1;
        while (a) {
            if (a & 1) res *= v;
            v *= v;
            a >>= 1;
        }
        return res;
    }
    template <typename Sc>
    void scan(Sc& a) {
        ll x;
        a.scan(x);
        *this = x;
    }
    template <typename Pr>
    void print(Pr& a) const {
        a.print(val);
    }
    template <typename Pr>
    void debug(Pr& a) const {
        a.print(val);
    }
};
template <typename T, int id>
T DynamicModInt<T, id>::mod = 998244353;
template <int id>
using dynamic_modint = DynamicModInt<unsigned int, id>;
using modint = dynamic_modint<-1>;
/**
 * @brief ModInt
 */
#line 4 "math/modular/montgomery-modint.hpp"

template <typename T>
struct MontgomeryReduction {
    static_assert(is_integral<T>::value, "template argument must be integral");
    static_assert(is_unsigned<T>::value, "template argument must be unsigned");

   private:
    using large_t = typename double_size_uint<T>::type;
    static constexpr int lg = numeric_limits<T>::digits;
    T mod;
    T r;
    T r2;
    T minv;
    T calc_inv() const {
        T t = 0, res = 0;
        rep(i, lg) {
            if (~t & 1) {
                t += mod;
                res += static_cast<T>(1) << i;
            }
            t >>= 1;
        }
        return res;
    }

   public:
    MontgomeryReduction(T x) { set_mod(x); }
    static constexpr int get_lg() { return lg; }
    void set_mod(T x) {
        assert(x > 0);
        assert(x & 1);
        assert(x <= INF<T>);
        mod = x;
        r = (-static_cast<T>(mod)) % mod;
        r2 = (-static_cast<large_t>(mod)) % mod;
        minv = calc_inv();
    }
    inline T get_r() const { return r; }
    inline T get_mod() const { return mod; }
    T reduce(large_t x) const {
        large_t tmp = (x + static_cast<large_t>(static_cast<T>(x) * minv) * mod) >> lg;
        return tmp >= mod ? tmp - mod : tmp;
    }
    T transform(large_t x) const { return reduce(x * r2); }
};
template <typename T, int id>
struct MontgomeryModInt : internal::modint_base {
    static_assert(is_integral<T>::value, "template argument must be integral");
    static_assert(is_unsigned<T>::value, "template argument must be unsigned");

   private:
    using large_t = typename double_size_uint<T>::type;
    T val;
    static MontgomeryReduction<T> reduction;

   public:
    MontgomeryModInt() : val(0) {}
    template <typename U, typename enable_if<is_integral<U>::value && is_unsigned<U>::value>::type* = nullptr>
    MontgomeryModInt(U x) : val(reduction.transform(x < (static_cast<large_t>(reduction.get_mod()) << reduction.get_lg()) ? static_cast<large_t>(x) : static_cast<large_t>(x % reduction.get_mod()))) {}
    template <typename U, typename enable_if<is_integral<U>::value && is_signed<U>::value>::type* = nullptr>
    MontgomeryModInt(U x) : MontgomeryModInt(static_cast<typename std::make_unsigned<U>::type>(x < 0 ? -x : x)) {
        if (x < 0 && val) val = reduction.get_mod() - val;
    }
    T get() const { return reduction.reduce(val); }
    static T get_mod() { return reduction.get_mod(); }
    static void set_mod(T x) { reduction.set_mod(x); }
    MontgomeryModInt& operator++() {
        val += reduction.get_r();
        if (val >= reduction.get_mod()) val -= reduction.get_mod();
        return *this;
    }
    MontgomeryModInt operator++(int) {
        MontgomeryModInt res = *this;
        ++*this;
        return res;
    }
    MontgomeryModInt& operator--() {
        if (val < reduction.get_r()) val += reduction.get_mod();
        val -= reduction.get_r();
        return *this;
    }
    MontgomeryModInt operator--(int) {
        MontgomeryModInt res = *this;
        --*this;
        return res;
    }
    MontgomeryModInt& operator+=(const MontgomeryModInt& r) {
        val += r.val;
        if (val >= reduction.get_mod()) val -= reduction.get_mod();
        return *this;
    }
    MontgomeryModInt& operator-=(const MontgomeryModInt& r) {
        if (val < r.val) val += reduction.get_mod();
        val -= r.val;
        return *this;
    }
    MontgomeryModInt& operator*=(const MontgomeryModInt& r) {
        val = reduction.reduce(static_cast<large_t>(val) * r.val);
        return *this;
    }
    MontgomeryModInt pow(ull n) const {
        MontgomeryModInt res = 1, tmp = *this;
        while (n) {
            if (n & 1) res *= tmp;
            tmp *= tmp;
            n >>= 1;
        }
        return res;
    }
    MontgomeryModInt inv() const { return pow(reduction.get_mod() - 2); }
    MontgomeryModInt& operator/=(const MontgomeryModInt& r) { return *this *= r.inv(); }
    friend MontgomeryModInt operator+(const MontgomeryModInt& l, const MontgomeryModInt& r) { return MontgomeryModInt(l) += r; }
    friend MontgomeryModInt operator-(const MontgomeryModInt& l, const MontgomeryModInt& r) { return MontgomeryModInt(l) -= r; }
    friend MontgomeryModInt operator*(const MontgomeryModInt& l, const MontgomeryModInt& r) { return MontgomeryModInt(l) *= r; }
    friend MontgomeryModInt operator/(const MontgomeryModInt& l, const MontgomeryModInt& r) { return MontgomeryModInt(l) /= r; }
    friend bool operator==(const MontgomeryModInt& l, const MontgomeryModInt& r) { return l.val == r.val; }
    friend bool operator!=(const MontgomeryModInt& l, const MontgomeryModInt& r) { return l.val != r.val; }
    template <typename Sc>
    void scan(Sc& a) {
        ll x;
        a.scan(x);
        *this = x;
    }
    template <typename Pr>
    void print(Pr& a) const {
        a.print(get());
    }
    template <typename Pr>
    void debug(Pr& a) const {
        a.print(get());
    }
};
template <typename T, int id>
MontgomeryReduction<T>
    MontgomeryModInt<T, id>::reduction = MontgomeryReduction<T>(998244353);
using ArbitraryModInt = MontgomeryModInt<unsigned int, -1>;
/**
 * @brief MontgomeryModInt(モンゴメリ乗算)
 */
#line 4 "math/number/miller-rabin.hpp"

template <typename T>
constexpr bool miller_rabin(ull n, const ull base[], int sz) {
    if (T::get_mod() != n) T::set_mod(n);
    ull d = n - 1;
    while (~d & 1) d >>= 1;
    const T e1 = 1, e2 = n - 1;
    rep(i, sz) {
        ull a = base[i];
        if (n <= a) return true;
        ull t = d;
        T y = T(a).pow(t);
        while (t != n - 1 && y != e1 && y != e2) {
            y *= y;
            t <<= 1;
        }
        if (y != e2 && (~t & 1)) return false;
    }
    return true;
}
constexpr bool is_prime_fast(ull n) {
    constexpr ull base_int[3] = {2, 7, 61}, base_ll[7] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
    if (n == 2) return true;
    if (n < 2 || n % 2 == 0) return false;
    if (n < (1u << 31)) return miller_rabin<MontgomeryModInt<unsigned int, -2>>(n, base_int, 3);
    return miller_rabin<MontgomeryModInt<ull, -2>>(n, base_ll, 7);
}
template <ull n>
constexpr bool is_prime_v = is_prime(n);
/**
 * @brief Miller-Rabin Primality Test(ミラーラビン素数判定)
 */
#line 3 "others/random.hpp"

template <typename Engine>
struct Random {
   private:
    Engine rnd;

   public:
    using result_type = typename Engine::result_type;
    Random() : Random(random_device{}()) {}
    Random(result_type seed) : rnd(seed) {}
    result_type operator()() { return rnd(); }
    template <typename IntType = ll>
    IntType uniform(IntType l, IntType r) {
        static_assert(is_integral<IntType>::value, "template argument must be an integral type");
        return uniform_int_distribution<IntType>{l, r}(rnd);
    }
    template <typename RealType = double>
    RealType uniform_real(RealType l, RealType r) {
        static_assert(is_floating_point<RealType>::value, "template argument must be a floating point type");
        return uniform_real_distribution<RealType>{l, r}(rnd);
    }
    bool uniform_bool() { return uniform<int>(0, 1); }
    template <typename T = ll>
    pair<T, T> uniform_pair(T l, T r) {
        T a, b;
        do {
            a = uniform<T>(l, r);
            b = uniform<T>(l, r);
        } while (a == b);
        if (a > b) swap(a, b);
        return {a, b};
    }
    template <typename Iter>
    void shuffle(const Iter& first, const Iter& last) {
        std::shuffle(first, last, rnd);
    }
    template <class T>
    vector<T> permutalion(T n) {
        static_assert(is_integral<T>::value, "template argument must be an integral type");
        vector<T> res(n);
        iota(res.begin(), res.end(), T());
        shuffle(all(res));
        return res;
    }
};
using Random32 = Random<mt19937>;
using Random64 = Random<mt19937_64>;
Random32 rand32;
Random64 rand64;
/**
 * @brief Random(乱数)
 */
#line 3 "string/run-length.hpp"

template <typename Cont, typename Comp>
vector<pair<typename Cont::value_type, int>> run_length(const Cont& c, const Comp& cmp) {
    vector<pair<typename Cont::value_type, int>> ret;
    if (c.empty()) return ret;
    ret.emplace_back(c.front(), 1);
    for (int i = 1; i < (int)c.size(); i++) {
        if (cmp(c[i], ret.back().first)) {
            ret.back().second++;
        } else {
            ret.emplace_back(c[i], 1);
        }
    }
    return ret;
}
template <typename Cont>
vector<pair<typename Cont::value_type, int>> run_length(const Cont& c) { return run_length(c, equal_to<typename Cont::value_type>()); }
#line 7 "math/number/pollard-rho.hpp"

template <typename T, typename Rand>
ull pollard_rho(ull n, Rand& rand) {
    if (~n & 1) return 2;
    if (T::get_mod() != n) T::set_mod(n);
    T c, e = 1;
    auto f = [&](T x) -> T { return x * x + c; };
    constexpr int m = 128;
    while (1) {
        c = rand.uniform(1ull, n - 1);
        T x = rand.uniform(2ull, n - 1), y = x;
        ull g = 1;
        while (g == 1) {
            T p = e, tx = x, ty = y;
            rep(i, m) {
                x = f(x);
                y = f(f(y));
                p *= x - y;
            }
            g = gcd(p.get(), n);
            if (g == 1) continue;
            rep(i, m) {
                tx = f(tx);
                ty = f(f(ty));
                g = gcd((tx - ty).get(), n);
                if (g != 1) {
                    if (g != n) return g;
                    break;
                }
            }
        }
    }
    return -1;
}
template <typename T = MontgomeryModInt<ull, -3>, typename Rand = Random64>
vector<ull> factorize(ull n, Rand& rand = rand64) {
    if (n == 1) return {};
    vector<ull> res;
    vector<ull> st = {n};
    while (!st.empty()) {
        ull t = st.back();
        st.pop_back();
        if (t == 1) continue;
        if (is_prime_fast(t)) {
            res.push_back(t);
            continue;
        }
        ull p = pollard_rho<T>(t, rand);
        st.push_back(p);
        st.push_back(t / p);
    }
    sort(all(res));
    return res;
}
template <typename T = MontgomeryModInt<ull, -3>, typename Rand = Random64>
vector<pair<ull, int>> expfactorize(ull n, Rand& rand = rand64) {
    auto res = factorize<T>(n, rand);
    return run_length(res);
}
/**
 * @brief Pollard's Rho Factorization(ポラード・ロー法)
 */
#line 6 "math/number/primitive-root.hpp"

template <typename T = MontgomeryModInt<ull, -4>, typename Rand = Random64>
ull primitive_root(ull n, Rand rand = rand64) {
    assert(is_prime_fast(n));
    if (n == 2) return 1;
    if (T::get_mod() != n) T::set_mod(n);
    auto divs = factorize(n - 1);
    divs.erase(unique(divs.begin(), divs.end()), divs.end());
    for (auto& x : divs) x = (n - 1) / x;
    const T e = 1;
    while (1) {
        ull g = rand.uniform(2ull, n - 1);
        bool ok = 1;
        for (auto x : divs) {
            if (T(g).pow(x) == e) {
                ok = false;
                break;
            }
        }
        if (ok) return g;
    }
}
template <ull p, enable_if_t<is_prime_v<p>>* = nullptr>
constexpr ull constexpr_primitive_root() {
    if (p == 2) return 1;
    if (p == 167772161) return 3;
    if (p == 469762049) return 3;
    if (p == 754974721) return 11;
    if (p == 998244353) return 3;
    if (p == 1224736769) return 3;
    if (p == 1811939329) return 11;
    if (p == 2013265921) return 11;
    rep(g, 2, p) {
        if (mod_pow(g, (p - 1) >> 1, p) != 1) return g;
    }
    return -1;
}
/**
 * @brief Primitive Root(原始根)
 */
#line 6 "math/convolution/convolution.hpp"

template <unsigned int p>
struct NthRoot {
   private:
    static constexpr unsigned int lg = msb((p - 1) & (1 - p));

   public:
    array<ModInt<p>, lg + 1> root, inv_root;
    array<ModInt<p>, max(0u, lg - 1)> rate2, irate2;
    array<ModInt<p>, max(0u, lg - 2)> rate3, irate3;
    constexpr NthRoot() : root{}, inv_root{} {
        root[lg] = mod_pow(constexpr_primitive_root<p>(), (p - 1) >> lg, p);
        inv_root[lg] = root[lg].pow(p - 2);
        ;
        rrep(i, lg) {
            root[i] = root[i + 1] * root[i + 1];
            inv_root[i] = inv_root[i + 1] * inv_root[i + 1];
        }
        {
            ModInt<p> prod = 1, iprod = 1;
            rep(i, lg - 1) {
                rate2[i] = root[i + 2] * prod;
                irate2[i] = inv_root[i + 2] * iprod;
                prod *= inv_root[i + 2];
                iprod *= root[i + 2];
            }
        }
        {
            ModInt<p> prod = 1, iprod = 1;
            rep(i, lg - 2) {
                rate3[i] = root[i + 3] * prod;
                irate3[i] = inv_root[i + 3] * iprod;
                prod *= inv_root[i + 3];
                iprod *= root[i + 3];
            }
        }
    }
    static constexpr unsigned int get_lg() { return lg; }
};
template <unsigned int p>
constexpr NthRoot<p> nth_root;
template <typename T, enable_if_t<is_modint<T>::value>* = nullptr>
void ntt(vector<T>& a) {
    constexpr unsigned int p = T::get_mod();
    const int sz = a.size();
    assert((unsigned int)sz <= ((1 - p) & (p - 1)));
    assert((sz & (sz - 1)) == 0);
    const int lg = msb(sz);
    static constexpr T im = nth_root<p>.root[2];
    for (int i = lg; i >= 1; i -= 2) {
        if (i == 1) {
            T z = 1;
            for (int j = 0; j < sz; j += (1 << i)) {
                for (int k = j; k < j + (1 << (i - 1)); ++k) {
                    const T x = a[k], y = a[k + (1 << (i - 1))] * z;
                    a[k] = x + y, a[k + (1 << (i - 1))] = x - y;
                }
                if (j + (1 << i) != sz) z *= nth_root<p>.rate2[__builtin_ctz(~(unsigned int)(j >> i))];
            }
        } else {
            const int offset = 1 << (i - 2);
            T z = 1;
            for (int j = 0; j < sz; j += (1 << i)) {
                for (int k = j; k < j + (1 << (i - 2)); ++k) {
                    const T z2 = z * z, z3 = z2 * z;
                    const T c0 = a[k], c1 = a[k + offset] * z, c2 = a[k + offset * 2] * z2, c3 = a[k + offset * 3] * z3;
                    const T c0c2 = c0 + c2, c0mc2 = c0 - c2, c1c3 = c1 + c3, c1mc3im = (c1 - c3) * im;
                    a[k] = c0c2 + c1c3;
                    a[k + offset] = c0c2 - c1c3;
                    a[k + offset * 2] = c0mc2 + c1mc3im;
                    a[k + offset * 3] = c0mc2 - c1mc3im;
                }
                if (j + (1 << i) != sz) z *= nth_root<p>.rate3[__builtin_ctz(~(unsigned int)(j >> i))];
            }
        }
    }
}
template <typename T, enable_if_t<is_modint<T>::value>* = nullptr>
void intt(vector<T>& a, const bool& f = true) {
    constexpr unsigned int p = T::get_mod();
    const int sz = a.size();
    assert((unsigned int)sz <= ((1 - p) & (p - 1)));
    assert((sz & (sz - 1)) == 0);
    const int lg = msb(sz);
    static constexpr T im = nth_root<p>.inv_root[2];
    for (int i = 2 - (lg & 1); i <= lg; i += 2) {
        if (i == 1) {
            T z = 1;
            for (int j = 0; j < sz; j += (1 << i)) {
                for (int k = j; k < j + (1 << (i - 1)); ++k) {
                    const T x = a[k], y = a[k + (1u << (i - 1))];
                    a[k] = x + y, a[k + (1u << (i - 1))] = (x - y) * z;
                }
                if (j + (1 << i) != sz) z *= nth_root<p>.irate2[__builtin_ctz(~(unsigned int)(j >> i))];
            }
        } else {
            const int offset = 1 << (i - 2);
            T z = 1;
            for (int j = 0; j < sz; j += (1 << i)) {
                for (int k = j; k < j + (1 << (i - 2)); ++k) {
                    const T z2 = z * z, z3 = z2 * z;
                    const T c0 = a[k], c1 = a[k + offset], c2 = a[k + offset * 2], c3 = a[k + offset * 3];
                    const T c0c1 = c0 + c1, c0mc1 = c0 - c1, c2c3 = c2 + c3, c2mc3im = (c2 - c3) * im;
                    a[k] = c0c1 + c2c3;
                    a[k + offset] = (c0mc1 + c2mc3im) * z;
                    a[k + offset * 2] = (c0c1 - c2c3) * z2;
                    a[k + offset * 3] = (c0mc1 - c2mc3im) * z3;
                }
                if (j + (1 << i) != sz) z *= nth_root<p>.irate3[__builtin_ctz(~(unsigned int)(j >> i))];
            }
        }
    }
    if (f) {
        const T inv_sz = T(1) / sz;
        for (auto& x : a) x *= inv_sz;
    }
}
template <typename T>
vector<T> convolution_naive(const vector<T>& a, const vector<T>& b) {
    const int sz1 = a.size(), sz2 = b.size();
    vector<T> c(sz1 + sz2 - 1);
    rep(i, sz1) rep(j, sz2) c[i + j] += a[i] * b[j];
    return c;
}
template <unsigned int p>
vector<ModInt<p>> convolution_for_any_mod(const vector<ModInt<p>>& a, const vector<ModInt<p>>& b);
template <typename T, enable_if_t<is_modint<T>::value>* = nullptr>
vector<T> convole(vector<T> a, vector<T> b) {
    const int n = a.size() + b.size() - 1;
    const int lg = ceil_log2(n);
    const int sz = 1 << lg;
    a.resize(sz), b.resize(sz);
    ntt(a), ntt(b);
    rep(i, sz) a[i] *= b[i];
    intt(a);
    a.resize(n);
    return a;
}
template <typename T, enable_if_t<is_modint<T>::value>* = nullptr>
vector<T> convolution(const vector<T>& a, const vector<T>& b) {
    constexpr unsigned int p = T::get_mod();
    const unsigned int sz1 = a.size(), sz2 = b.size();
    if (sz1 == 0 || sz2 == 0) return {};
    if (sz1 <= 64 || sz2 <= 64) return convolution_naive(a, b);
    if constexpr (((p - 1) & (1 - p)) >= 128) {
        if (sz1 + sz2 - 1 <= ((p - 1) & (1 - p))) return convole(a, b);
    }
    return convolution_for_any_mod(a, b);
}

template <unsigned int p = 998244353>
vector<ll> convolution(const vector<ll>& a, const vector<ll>& b) {
    const int sz1 = a.size(), sz2 = b.size();
    vector<ModInt<p>> a1(sz1), b1(sz2);
    rep(i, sz1) a1[i] = a[i];
    rep(i, sz2) b1[i] = b[i];
    auto c1 = convolution(a1, b1);
    vector<ll> c(sz1 + sz2 - 1);
    rep(i, sz1 + sz2 - 1) c[i] = c1[i].get();
    return c;
}
template <unsigned int p>
vector<ModInt<p>> convolution_for_any_mod(const vector<ModInt<p>>& a, const vector<ModInt<p>>& b) {
    const int sz1 = a.size(), sz2 = b.size();
    assert(sz1 + sz2 - 1 <= (1 << 26));
    vector<ll> a1(sz1), b1(sz2);
    rep(i, sz1) a1[i] = a[i].get();
    rep(i, sz2) b1[i] = b[i].get();
    static constexpr ull MOD1 = 469762049;
    static constexpr ull MOD2 = 1811939329;
    static constexpr ull MOD3 = 2013265921;
    static constexpr ull INV1_2 = mod_pow(MOD1, MOD2 - 2, MOD2);
    static constexpr ull INV1_3 = mod_pow(MOD1, MOD3 - 2, MOD3);
    static constexpr ull INV2_3 = mod_pow(MOD2, MOD3 - 2, MOD3);
    auto c1 = convolution<MOD1>(a1, b1);
    auto c2 = convolution<MOD2>(a1, b1);
    auto c3 = convolution<MOD3>(a1, b1);
    vector<ModInt<p>> c(sz1 + sz2 - 1);
    rep(i, sz1 + sz2 - 1) {
        ull x1 = c1[i];
        ull x2 = (c2[i] - x1 + MOD2) * INV1_2 % MOD2;
        ull x3 = ((c3[i] - x1 + MOD3) * INV1_3 % MOD3 - x2 + MOD3) * INV2_3 % MOD3;
        c[i] = ModInt<p>(x1 + (x2 + x3 * MOD2) % p * MOD1);
    }
    return c;
}
/**
 * @brief Convolution(畳み込み)
 */
#line 4 "math/others/combinatorics.hpp"

template <typename T>
struct Combinatorics {
   private:
    static vector<T> dat, idat;

   public:
    static void extend(int sz) {
        const int pre_sz = dat.size();
        if (sz < pre_sz) return;
        dat.resize(sz + 1, 1);
        idat.resize(sz + 1, 1);
        for (int i = pre_sz; i <= sz; i++) dat[i] = dat[i - 1] * i;
        idat[sz] = T(1) / dat[sz];
        for (int i = sz - 1; i >= pre_sz; i--) idat[i] = idat[i + 1] * (i + 1);
    }
    static T fac(ll n) {
        if (n < 0) return T();
        extend(n);
        return dat[n];
    }
    static T finv(ll n) {
        if (n < 0) return T();
        extend(n);
        return idat[n];
    }
    static T inv(ll n) {
        if (n <= 0) return T();
        extend(n);
        return dat[n - 1] * idat[n];
    }
    static T com(ll n, ll k) {
        if (k < 0 || n < k) return T();
        extend(n);
        return dat[n] * idat[k] * idat[n - k];
    }
    static T hom(ll n, ll k) {
        if (n < 0 || k < 0) return T();
        return k == 0 ? 1 : com(n + k - 1, k);
    }
    static inline T per(ll n, ll k) {
        if (k < 0 || n < k) return T();
        extend(n);
        return dat[n] * idat[n - k];
    }
};
template <typename T>
vector<T> Combinatorics<T>::dat = vector<T>(2, 1);
template <typename T>
vector<T> Combinatorics<T>::idat = vector<T>(2, 1);
template <long long p>
struct COMB {
   private:
    static vector<vector<ModInt<p>>> comb;
    static void init() {
        if (!comb.empty()) return;
        comb.assign(p, vector<ModInt<p>>(p));
        comb[0][0] = 1;
        for (int i = 1; i < p; i++) {
            comb[i][0] = 1;
            for (int j = i; j > 0; j--) comb[i][j] = comb[i - 1][j - 1] + comb[i - 1][j];
        }
    }

   public:
    COMB() {
        init();
    }
    ModInt<p> com(int n, int k) {
        init();
        ModInt<p> ret = 1;
        while (n > 0 || k > 0) {
            int ni = n % p, ki = k % p;
            ret *= comb[ni][ki];
            n /= p;
            k /= p;
        }
        return ret;
    }
};
template <long long p>
vector<vector<ModInt<p>>> COMB<p>::comb = vector<vector<ModInt<p>>>();
/**
 * @brief Combinatorics(組み合わせ)
 */
#line 5 "math/fps/fps.hpp"

template <typename mint = ModInt<998244353>>
struct FormalPowerSeries : vector<mint> {
    using vector<mint>::vector;
    using FPS = FormalPowerSeries<mint>;
    using Comb = Combinatorics<mint>;

   private:
    static constexpr unsigned int p = mint::get_mod();

   public:
    FormalPowerSeries() : vector<mint>() {}
    FormalPowerSeries(const vector<mint>& v) : vector<mint>(v) {}
    FormalPowerSeries(vector<mint>&& v) : vector<mint>(move(v)) {}
    inline void shrink() {
        while (!(*this).empty() && (*this).back() == mint()) (*this).pop_back();
    }
    FPS& dot(const FPS& r) {
        rep(i, min((*this).size(), r.size()))(*this)[i] *= r[i];
        return *this;
    }
    FPS inv(int d = -1) const {
        const int n = (*this).size();
        if (d == -1) d = n;
        FPS res(d);
        res[0] = (*this)[0].inv();
        for (int sz = 1; sz < d; sz <<= 1) {
            FPS f((*this).begin(), (*this).begin() + min(n, 2 * sz));
            FPS g(res.begin(), res.begin() + sz);
            f.resize(2 * sz), g.resize(2 * sz);
            ntt(f), ntt(g);
            f.dot(g);
            intt(f);
            rep(i, sz) f[i] = 0;
            ntt(f);
            f.dot(g);
            intt(f);
            rep(j, sz, min(2 * sz, d)) res[j] = -f[j];
        }
        return res;
    }
    FPS operator+() const { return *this; }
    FPS operator-() const {
        FPS res(*this);
        for (auto& x : res) x = -x;
        return res;
    }
    FPS& operator+=(const mint& r) {
        shrink();
        if ((*this).empty()) (*this).resize(1);
        (*this)[0] += r;
        return *this;
    }
    FPS& operator-=(const mint& r) {
        shrink();
        if ((*this).empty()) (*this).resize(1);
        (*this)[0] -= r;
        return *this;
    }
    FPS& operator*=(const mint& r) {
        shrink();
        for (auto& x : *this) x *= r;
        return *this;
    }
    FPS& operator/=(const mint& r) {
        shrink()(*this) *= r.inv();
        return *this;
    }
    FPS& operator+=(const FPS& r) {
        shrink();
        if ((*this).size() < r.size()) (*this).resize(r.size());
        rep(i, r.size())(*this)[i] += r[i];
        return *this;
    }
    FPS& operator-=(const FPS& r) {
        shrink();
        if ((*this).size() < r.size()) (*this).resize(r.size());
        rep(i, r.size())(*this)[i] -= r[i];
        return *this;
    }
    FPS& operator*=(const FPS& r) {
        shrink();
        auto ret = convolution(*this, r);
        (*this) = {ret.begin(), ret.end()};
        return *this;
    }
    FPS& operator/=(FPS r) {
        shrink();
        const int n = (*this).size(), m = r.size();
        if (n < m) {
            (*this).clear();
            return *this;
        }
        const int d = n - m + 1;
        reverse((*this).begin(), (*this).end());
        reverse(r.begin(), r.end());
        (*this).resize(d);
        (*this) *= r.inv(d);
        (*this).resize(d);
        reverse((*this).begin(), (*this).end());
        return *this;
    }
    FPS& operator%=(const FPS& r) {
        shrink();
        const int n = (*this).size(), m = r.size();
        if (n < m) return *this;
        (*this) -= (*this) / r * r;
        shrink();
        return *this;
    }
    FPS& operator<<=(ll k) {
        shrink();
        (*this).insert((*this).begin(), k, mint(0));
        return *this;
    }
    FPS& operator>>=(ll k) {
        shrink();
        if (k > (ll)(*this).size())
            (*this).clear();
        else
            (*this).erase((*this).begin(), (*this).begin() + k);
        return *this;
    }
    FPS operator<<(ll k) const { return FPS(*this) <<= k; }
    FPS operator>>(ll k) const { return FPS(*this) >>= k; }
    friend FPS operator+(const FPS& l, const mint& r) { return FPS(l) += r; }
    friend FPS operator-(const FPS& l, const mint& r) { return FPS(l) -= r; }
    friend FPS operator*(const FPS& l, const mint& r) { return FPS(l) *= r; }
    friend FPS operator/(const FPS& l, const mint& r) { return FPS(l) /= r; }
    friend FPS operator+(const mint& l, const FPS& r) { return FPS(r) += l; }
    friend FPS operator-(const mint& l, const FPS& r) { return FPS(-r) += l; }
    friend FPS operator*(const mint& l, const FPS& r) { return FPS(r) *= l; }
    friend FPS operator+(const FPS& l, const FPS& r) { return FPS(l) += r; }
    friend FPS operator-(const FPS& l, const FPS& r) { return FPS(l) -= r; }
    friend FPS operator*(const FPS& l, const FPS& r) { return FPS(l) *= r; }
    friend FPS operator/(const FPS& l, const FPS& r) { return FPS(l) /= r; }
    friend FPS operator%(const FPS& l, const FPS& r) { return FPS(l) %= r; }
    pair<FPS, FPS> div_mod(const FPS& r) const {
        FPS q = (*this) / r;
        FPS m;
        if ((*this).size() >= r.size())
            m = (*this) - q * r;
        else
            m = *this;
        q.shrink(), m.shrink();
        return {q, m};
    }
    mint operator()(const mint& x) const {
        mint res = 0, w = 1;
        for (auto& v : *this) res += v * w, w *= x;
        return res;
    }
    FPS diff() const {
        const int n = (*this).size();
        FPS res(n - 1);
        rep(i, 1, n) res[i - 1] = (*this)[i] * i;
        return res;
    }
    FPS& inplace_diff() {
        shrink();
        (*this).erase((*this).begin());
        mint coeff = 1;
        for (int i = 0; i < (int)(*this).size(); i++) {
            (*this)[i] *= coeff;
            coeff++;
        }
        return *this;
    }
    FPS integral() const {
        const int n = (*this).size();
        FPS res(n + 1);
        Comb::extend(n);
        rep(i, n) res[i + 1] = (*this)[i] * Comb::inv(i + 1);
        return res;
    }
    FPS& inplace_integral() {
        shrink();
        const int n = (*this).size();
        vector<mint> iv(n + 1, 1);
        rep(i, 2, n + 1) iv[i] = -iv[p % i] * (p / i);
        (*this).insert((*this).begin(), mint(0));
        rep(i, 1, n + 1)(*this)[i] *= iv[i];
        return *this;
    }
    FPS log(int d = -1) const {
        const int n = (*this).size();
        if (d == -1) d = n;
        FPS res = diff() * inv(d);
        res.resize(d - 1);
        return res.integral();
    }
    FPS& inplace_log(int d = -1) {
        shrink();
        const int n = (*this).size();
        if (d == -1) d = n;
        FPS tmp = inv(d);
        (*this).inplace_diff() *= tmp;
        (*this).resize(d - 1);
        return (*this).inplace_integral();
    }
    FPS exp(int d = -1) const {
        const int n = (*this).size();
        if (d == -1) d = n;
        if (n <= 1) {
            FPS res(d, mint());
            res[0] = 1;
            return res;
        }
        FPS f = {mint(1) + (*this)[0], (*this)[1]}, res{1, (*this)[1]};
        for (int sz = 2; sz < d; sz <<= 1) {
            f.insert(f.end(), (*this).begin() + min(sz, n), (*this).begin() + min(n, sz << 1));
            f.resize(sz << 1);
            res = res * (f - res.log(sz << 1));
            res.resize(sz << 1);
        }
        res.resize(d);
        return res;
    }
    FPS pow(ll k, int d = -1) const {
        const int n = (*this).size();
        if (d == -1) d = n;
        if (k == 0) {
            FPS ans(d, mint());
            ans[0] = 1;
            return ans;
        }
        for (int i = 0; i < n; i++) {
            if ((*this)[i] != mint()) {
                if (i > d / k) return FPS(d, mint());
                mint rev = (*this)[i].inv();
                FPS res = (((*this * rev) >> i).log(d) * k).exp(d) * ((*this)[i].pow(k));
                res = (res << (i * k));
                res.resize(d);
                return res;
            }
        }
        return FPS(d, mint());
    }
    FPS sqrt(
        const function<mint(mint)>& get_sqrt = [](mint) { return mint(1); }, int d = -1) const {
        const int n = (*this).size();
        if (d == -1) d = n;
        if ((*this)[0] == mint(0)) {
            rep(i, 1, n) {
                if ((*this)[i] != mint(0)) {
                    if (i & 1) return {};
                    if (d - i / 2 <= 0) break;
                    auto res = (*this >> i).sqrt(get_sqrt, d - i / 2);
                    if (res.empty()) return {};
                    res = res << (i / 2);
                    res.resize(d);
                    return res;
                }
            }
            return FPS(d);
        }
        auto sqr = get_sqrt((*this)[0]);
        if (sqr * sqr != (*this)[0]) return {};
        FPS res{sqr};
        const mint inv2 = mint(2).inv();
        FPS f = {(*this)[0]};
        for (int i = 1; i < d; i <<= 1) {
            if (i < n) f.insert(f.end(), (*this).begin() + i, (*this).begin() + min(n, i << 1));
            if ((int)f.size() < (i << 1)) f.resize(i << 1);
            res = (res + f * res.inv(i << 1)) * inv2;
        }
        res.resize(d);
        return res;
    }
};
/**
 * @brief Formal Power Series(形式的冪級数)
 */
#line 3 "math/fps/multipoint-evaluation-geometric.hpp"

template <typename T>
vector<T> multipoint_evaluation_geometric(FormalPowerSeries<T> f, T a, T r, int m) {
    if (m == 0) return {};
    const int n = f.size();
    if (r == T(0)) {
        vector<T> res(m, f[0]);
        res[0] = f(a);
        return res;
    }
    if (n < 64 || m < 64) {
        vector<T> res(m);
        rep(i, m) res[i] = f(a), a *= r;
        return res;
    }
    T pow_a = 1;
    rep(i, n) f[i] *= pow_a, pow_a *= a;
    auto calc = [](T r, int m) -> vector<T> {
        vector<T> res(m);
        T pow_r = 1;
        res[0] = 1;
        rep(i, m - 1) res[i + 1] = res[i] * pow_r, pow_r *= r;
        return res;
    };
    auto r_memo = calc(r, n + m - 1), r_inv_memo = calc(T(1) / r, max(n, m));
    rep(i, n) f[i] *= r_inv_memo[i];
    reverse(f.begin(), f.end());
    const int l = 1 << ceil_log2(n + m - 1);
    f.resize(l);
    r_memo.resize(l);
    ntt(f), ntt(r_memo);
    rep(i, l) f[i] *= r_memo[i];
    intt(f);
    f >>= n - 1;
    f.resize(m);
    rep(i, m) f[i] *= r_inv_memo[i];
    return f;
}
/**
 * @brief Multipoint Evaluation (Geometric Sequence)
 * @ref https://noshi91.github.io/algorithm-encyclopedia/chirp-z-transform
 */
#line 4 "math/fps/polynomial-interpolation-gemoetric.hpp"

template <typename T>
FormalPowerSeries<T> polynomial_interpolation_gemoetric(vector<T> y, const T& a, const T& r) {
    const int n = y.size();
    if (n == 0) return {};
    if (n == 1) return {y[0]};
    if (r == 0) {
        return {y[1], (y[0] - y[1]) / a};
    }
    vector<T> pow_r(n + 1), pow_invr(n + 1), inv_s(n);
    T s = 1;
    const T inv_r = T(1) / r;
    {
        pow_r[0] = pow_invr[0] = inv_s[0] = 1;
        rep(i, n) {
            if (i) s *= (1 - pow_r[i]);
            pow_r[i + 1] = pow_r[i] * r;
            pow_invr[i + 1] = pow_invr[i] * inv_r;
        }
        inv_s[n - 1] = T(1) / s;
        s *= (1 - pow_r[n]);
        rrep(i, n - 1) inv_s[i] = inv_s[i + 1] * (1 - pow_r[i + 1]);
    }
    T v = 1;
    rep(i, n) {
        y[i] *= v * inv_s[i] * inv_s[n - i - 1];
        if (i & 1) y[i] = -y[i];
        if (i != n - 1) v *= pow_invr[n - 2 - i];
    }
    FormalPowerSeries<T> f1(n + 1);
    v = 1;
    rep(i, n + 1) {
        if (i == 0 || i == n)
            f1[i] = v;
        else
            f1[i] = v * s * inv_s[i] * inv_s[n - i];
        if (i & 1) f1[i] = -f1[i];
        v *= pow_r[i];
    }
    FormalPowerSeries<T> f2 = multipoint_evaluation_geometric(FormalPowerSeries<T>(y), T(1), r, n);
    auto res = (f1 * f2);
    res.resize(n);
    reverse(res.begin(), res.end());
    const T inva = T(1) / a;
    v = 1;
    rep(i, n) res[i] *= v, v *= inva;
    return res;
}
/**
 * @brief Polynomial Interpolation (Geometric Sequence)
 * @ref https://noshi91.github.io/algorithm-encyclopedia/polynomial-interpolation-geometric
 */

Back to top page