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

#line 3 "library/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 "library/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 "library/template/type-traits.hpp"

#line 5 "library/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 "library/template/in.hpp"
#include <unistd.h>
#line 5 "library/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 "library/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 "library/template/func.hpp"

#line 8 "library/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>
T binary_gcd(T x_, T y_) {
    T x = x_ < 0 ? -x_ : x_, y = y_ < 0 ? -y_ : y_;
    if (!x || !y) return x + y;
    int n = __builtin_ctzll(x), m = __builtin_ctzll(y);
    x >>= n, y >>= m;
    while (x != y) {
        if (x > y) {
            x = (x - y) >> __builtin_ctzll(x - y);
        } else {
            y = (y - x) >> __builtin_ctzll(y - x);
        }
    }
    return x << std::min(n, m);
}
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 "library/template/util.hpp"

#line 6 "library/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 "library/template/template.hpp"
using namespace std;
#line 3 "library/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;
        if (a < 0) {
            a = -a;
            v = v.inv();
        }
        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;
        if (a < 0) {
            a = -a;
            v = v.inv();
        }
        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 "library/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 || n < 0) 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 4 "code.cpp"
using mint = ModInt<998244353>;
Combinatorics<mint> comb;
int main() {
    LL(n);
    if (n & 1) fin(0);
    n /= 2;
    n--;
    print(mint(2).pow(2 * n + 1) - comb.com(2 * n + 3, n + 1) + comb.com(2 * n + 1, n));
    return 0;
    /*
    rep(n, 1, 11) {
        ll sum = 0;
        rep(bit, 1 << (2 * n)) {
            stack<ll> st;
            ll ans = 0;
            bool f = true;
            rep(i, 2 * n) {
                if ((bit >> i) & 1) {
                    st.emplace(i);
                } else {
                    if (st.empty()) {
                        f = false;
                        break;
                    }
                    st.pop();
                    ans += st.size();
                }
            }
            f &= st.empty();
            if (f) sum += ans;
        }
        print(sum);
    }*/
}