// #pragma GCC target("avx2")
// #pragma GCC optimize("O3")
// #pragma GCC optimize("unroll-loops")
// #define INTERACTIVE

#include <bits/stdc++.h>
using namespace std;

namespace templates {
// type
using ll  = long long;
using ull = unsigned long long;
using Pii = pair<int, int>;
using Pil = pair<int, ll>;
using Pli = pair<ll, int>;
using Pll = pair<ll, ll>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using qp = priority_queue<T, vector<T>, greater<T>>;
// clang-format off
#define vec(T, A, ...) vector<T> A(__VA_ARGS__);
#define vvec(T, A, h, ...) vector<vector<T>> A(h, vector<T>(__VA_ARGS__));
#define vvvec(T, A, h1, h2, ...) vector<vector<vector<T>>> A(h1, vector<vector<T>>(h2, vector<T>(__VA_ARGS__)));
// clang-format on

// for loop
#define fori1(a) for (ll _ = 0; _ < (a); _++)
#define fori2(i, a) for (ll i = 0; i < (a); i++)
#define fori3(i, a, b) for (ll i = (a); i < (b); i++)
#define fori4(i, a, b, c) for (ll i = (a); ((c) > 0 || i > (b)) && ((c) < 0 || i < (b)); i += (c))
#define overload4(a, b, c, d, e, ...) e
#define fori(...) overload4(__VA_ARGS__, fori4, fori3, fori2, fori1)(__VA_ARGS__)

// declare and input
// clang-format off
#define INT(...) int __VA_ARGS__; inp(__VA_ARGS__);
#define LL(...) ll __VA_ARGS__; inp(__VA_ARGS__);
#define STRING(...) string __VA_ARGS__; inp(__VA_ARGS__);
#define CHAR(...) char __VA_ARGS__; inp(__VA_ARGS__);
#define DOUBLE(...) double __VA_ARGS__; STRING(str___); __VA_ARGS__ = stod(str___);
#define VEC(T, A, n) vector<T> A(n); inp(A);
#define VVEC(T, A, n, m) vector<vector<T>> A(n, vector<T>(m)); inp(A);
// clang-format on

// const value
const ll MOD1   = 1000000007;
const ll MOD9   = 998244353;
const double PI = acos(-1);

// other macro
#if !defined(RIN__LOCAL) && !defined(INTERACTIVE)
#define endl "\n"
#endif
#define spa ' '
#define len(A) ll(A.size())
#define all(A) begin(A), end(A)

// function
vector<char> stoc(string &S) {
    int n = S.size();
    vector<char> ret(n);
    for (int i = 0; i < n; i++) ret[i] = S[i];
    return ret;
}
string ctos(vector<char> &S) {
    int n      = S.size();
    string ret = "";
    for (int i = 0; i < n; i++) ret += S[i];
    return ret;
}

template <class T>
auto min(const T &a) {
    return *min_element(all(a));
}
template <class T>
auto max(const T &a) {
    return *max_element(all(a));
}
template <class T, class S>
auto clamp(T &a, const S &l, const S &r) {
    return (a > r ? r : a < l ? l : a);
}
template <class T, class S>
inline bool chmax(T &a, const S &b) {
    return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
    return (a > b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chclamp(T &a, const S &l, const S &r) {
    auto b = clamp(a, l, r);
    return (a != b ? a = b, 1 : 0);
}

template <typename T>
T sum(vector<T> &A) {
    T tot = 0;
    for (auto a : A) tot += a;
    return tot;
}

template <typename T>
vector<T> compression(vector<T> X) {
    sort(all(X));
    X.erase(unique(all(X)), X.end());
    return X;
}

// input and output
namespace io {
// __int128_t
std::istream &operator>>(std::istream &is, __int128_t &value) {
    std::string str;
    is >> str;
    value    = 0;
    int sign = 1;
    for (size_t i = 0; i < str.size(); i++) {
        if (i == 0 && str[i] == '-') {
            sign = -1;
            continue;
        }
        value = value * 10 + str[i] - '0';
    }
    value *= sign;
    return is;
}

std::ostream &operator<<(std::ostream &dest, __int128_t value) {
    std::ostream::sentry s(dest);
    if (s) {
        __uint128_t tmp = value < 0 ? -value : value;
        char buffer[128];
        char *d = std::end(buffer);
        do {
            --d;
            *d = "0123456789"[tmp % 10];
            tmp /= 10;
        } while (tmp != 0);
        if (value < 0) {
            --d;
            *d = '-';
        }
        int len = std::end(buffer) - d;
        if (dest.rdbuf()->sputn(d, len) != len) {
            dest.setstate(std::ios_base::badbit);
        }
    }
    return dest;
}

// vector<T>
template <typename T>
istream &operator>>(istream &is, vector<T> &A) {
    for (auto &a : A) is >> a;
    return is;
}
template <typename T>
ostream &operator<<(ostream &os, vector<T> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        os << A[i];
        if (i != A.size() - 1) os << ' ';
    }
    return os;
}

// vector<vector<T>>
template <typename T>
istream &operator>>(istream &is, vector<vector<T>> &A) {
    for (auto &a : A) is >> a;
    return is;
}
template <typename T>
ostream &operator<<(ostream &os, vector<vector<T>> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        os << A[i];
        if (i != A.size() - 1) os << endl;
    }
    return os;
}

// pair<S, T>
template <typename S, typename T>
istream &operator>>(istream &is, pair<S, T> &A) {
    is >> A.first >> A.second;
    return is;
}
template <typename S, typename T>
ostream &operator<<(ostream &os, pair<S, T> &A) {
    os << A.first << ' ' << A.second;
    return os;
}

// vector<pair<S, T>>
template <typename S, typename T>
istream &operator>>(istream &is, vector<pair<S, T>> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        is >> A[i];
    }
    return is;
}
template <typename S, typename T>
ostream &operator<<(ostream &os, vector<pair<S, T>> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        os << A[i];
        if (i != A.size() - 1) os << endl;
    }
    return os;
}

// tuple
template <typename T, size_t N>
struct TuplePrint {
    static ostream &print(ostream &os, const T &t) {
        TuplePrint<T, N - 1>::print(os, t);
        os << ' ' << get<N - 1>(t);
        return os;
    }
};
template <typename T>
struct TuplePrint<T, 1> {
    static ostream &print(ostream &os, const T &t) {
        os << get<0>(t);
        return os;
    }
};
template <typename... Args>
ostream &operator<<(ostream &os, const tuple<Args...> &t) {
    TuplePrint<decltype(t), sizeof...(Args)>::print(os, t);
    return os;
}

// io functions
void FLUSH() {
    cout << flush;
}

void print() {
    cout << endl;
}
template <class Head, class... Tail>
void print(Head &&head, Tail &&...tail) {
    cout << head;
    if (sizeof...(Tail)) cout << spa;
    print(std::forward<Tail>(tail)...);
}

template <typename T, typename S>
void prisep(vector<T> &A, S sep) {
    int n = A.size();
    for (int i = 0; i < n; i++) {
        cout << A[i];
        if (i != n - 1) cout << sep;
    }
    cout << endl;
}
template <typename T, typename S>
void priend(T A, S end) {
    cout << A << end;
}
template <typename T>
void prispa(T A) {
    priend(A, spa);
}
template <typename T, typename S>
bool printif(bool f, T A, S B) {
    if (f)
        print(A);
    else
        print(B);
    return f;
}

template <class... T>
void inp(T &...a) {
    (cin >> ... >> a);
}

} // namespace io
using namespace io;

// read graph
vector<vector<int>> read_edges(int n, int m, bool direct = false, int indexed = 1) {
    vector<vector<int>> edges(n, vector<int>());
    for (int i = 0; i < m; i++) {
        INT(u, v);
        u -= indexed;
        v -= indexed;
        edges[u].push_back(v);
        if (!direct) edges[v].push_back(u);
    }
    return edges;
}
vector<vector<int>> read_tree(int n, int indexed = 1) {
    return read_edges(n, n - 1, false, indexed);
}

template <typename T = long long>
vector<vector<pair<int, T>>> read_wedges(int n, int m, bool direct = false, int indexed = 1) {
    vector<vector<pair<int, T>>> edges(n, vector<pair<int, T>>());
    for (int i = 0; i < m; i++) {
        INT(u, v);
        T w;
        inp(w);
        u -= indexed;
        v -= indexed;
        edges[u].push_back({v, w});
        if (!direct) edges[v].push_back({u, w});
    }
    return edges;
}
template <typename T = long long>
vector<vector<pair<int, T>>> read_wtree(int n, int indexed = 1) {
    return read_wedges<T>(n, n - 1, false, indexed);
}

// yes / no
namespace yesno {

// yes
inline bool yes(bool f = true) {
    cout << (f ? "yes" : "no") << endl;
    return f;
}
inline bool Yes(bool f = true) {
    cout << (f ? "Yes" : "No") << endl;
    return f;
}
inline bool YES(bool f = true) {
    cout << (f ? "YES" : "NO") << endl;
    return f;
}

// no
inline bool no(bool f = true) {
    cout << (!f ? "yes" : "no") << endl;
    return f;
}
inline bool No(bool f = true) {
    cout << (!f ? "Yes" : "No") << endl;
    return f;
}
inline bool NO(bool f = true) {
    cout << (!f ? "YES" : "NO") << endl;
    return f;
}

// possible
inline bool possible(bool f = true) {
    cout << (f ? "possible" : "impossible") << endl;
    return f;
}
inline bool Possible(bool f = true) {
    cout << (f ? "Possible" : "Impossible") << endl;
    return f;
}
inline bool POSSIBLE(bool f = true) {
    cout << (f ? "POSSIBLE" : "IMPOSSIBLE") << endl;
    return f;
}

// impossible
inline bool impossible(bool f = true) {
    cout << (!f ? "possible" : "impossible") << endl;
    return f;
}
inline bool Impossible(bool f = true) {
    cout << (!f ? "Possible" : "Impossible") << endl;
    return f;
}
inline bool IMPOSSIBLE(bool f = true) {
    cout << (!f ? "POSSIBLE" : "IMPOSSIBLE") << endl;
    return f;
}

// Alice Bob
inline bool Alice(bool f = true) {
    cout << (f ? "Alice" : "Bob") << endl;
    return f;
}
inline bool Bob(bool f = true) {
    cout << (f ? "Bob" : "Alice") << endl;
    return f;
}

// Takahashi Aoki
inline bool Takahashi(bool f = true) {
    cout << (f ? "Takahashi" : "Aoki") << endl;
    return f;
}
inline bool Aoki(bool f = true) {
    cout << (f ? "Aoki" : "Takahashi") << endl;
    return f;
}

} // namespace yesno
using namespace yesno;

} // namespace templates
using namespace templates;

template <int MOD>
struct Modint {
    int x;
    Modint() : x(0) {}
    Modint(int64_t y) {
        if (y >= 0)
            x = y % MOD;
        else
            x = (y % MOD + MOD) % MOD;
    }

    Modint &operator+=(const Modint &p) {
        x += p.x;
        if (x >= MOD) x -= MOD;
        return *this;
    }

    Modint &operator-=(const Modint &p) {
        x -= p.x;
        if (x < 0) x += MOD;
        return *this;
    }

    Modint &operator*=(const Modint &p) {
        x = int(1LL * x * p.x % MOD);
        return *this;
    }

    Modint &operator/=(const Modint &p) {
        *this *= p.inverse();
        return *this;
    }

    Modint &operator%=(const Modint &p) {
        assert(p.x == 0);
        return *this;
    }

    Modint operator-() const {
        return Modint(-x);
    }

    Modint &operator++() {
        x++;
        if (x == MOD) x = 0;
        return *this;
    }

    Modint &operator--() {
        if (x == 0) x = MOD;
        x--;
        return *this;
    }

    Modint operator++(int) {
        Modint result = *this;
        ++*this;
        return result;
    }

    Modint operator--(int) {
        Modint result = *this;
        --*this;
        return result;
    }

    friend Modint operator+(const Modint &lhs, const Modint &rhs) {
        return Modint(lhs) += rhs;
    }

    friend Modint operator-(const Modint &lhs, const Modint &rhs) {
        return Modint(lhs) -= rhs;
    }

    friend Modint operator*(const Modint &lhs, const Modint &rhs) {
        return Modint(lhs) *= rhs;
    }

    friend Modint operator/(const Modint &lhs, const Modint &rhs) {
        return Modint(lhs) /= rhs;
    }

    friend Modint operator%(const Modint &lhs, const Modint &rhs) {
        assert(rhs.x == 0);
        return Modint(lhs);
    }

    bool operator==(const Modint &p) const {
        return x == p.x;
    }

    bool operator!=(const Modint &p) const {
        return x != p.x;
    }

    bool operator<(const Modint &rhs) const {
        return x < rhs.x;
    }

    bool operator<=(const Modint &rhs) const {
        return x <= rhs.x;
    }

    bool operator>(const Modint &rhs) const {
        return x > rhs.x;
    }

    bool operator>=(const Modint &rhs) const {
        return x >= rhs.x;
    }

    Modint inverse() const {
        int a = x, b = MOD, u = 1, v = 0, t;
        while (b > 0) {
            t = a / b;
            a -= t * b;
            u -= t * v;
            std::swap(a, b);
            std::swap(u, v);
        }
        return Modint(u);
    }

    Modint pow(int64_t k) const {
        Modint ret(1);
        Modint y(x);
        while (k > 0) {
            if (k & 1) ret *= y;
            y *= y;
            k >>= 1;
        }
        return ret;
    }

    std::pair<int, int> to_frac(int max_n = 1000) const {
        int y = x;
        for (int i = 1; i <= max_n; i++) {
            if (y <= max_n) {
                return {y, i};
            } else if (MOD - y <= max_n) {
                return {-(MOD - y), i};
            }
            y = (y + x) % MOD;
        }
        return {-1, -1};
    }

    friend std::ostream &operator<<(std::ostream &os, const Modint &p) {
        return os << p.x;
    }

    friend std::istream &operator>>(std::istream &is, Modint &p) {
        int64_t y;
        is >> y;
        p = Modint<MOD>(y);
        return (is);
    }

    static int get_mod() {
        return MOD;
    }
};

struct Arbitrary_Modint {
    int x;
    static int MOD;

    static void set_mod(int mod) {
        MOD = mod;
    }

    Arbitrary_Modint() : x(0) {}
    Arbitrary_Modint(int64_t y) {
        if (y >= 0)
            x = y % MOD;
        else
            x = (y % MOD + MOD) % MOD;
    }

    Arbitrary_Modint &operator+=(const Arbitrary_Modint &p) {
        x += p.x;
        if (x >= MOD) x -= MOD;
        return *this;
    }

    Arbitrary_Modint &operator-=(const Arbitrary_Modint &p) {
        x -= p.x;
        if (x < 0) x += MOD;
        return *this;
    }

    Arbitrary_Modint &operator*=(const Arbitrary_Modint &p) {
        x = int(1LL * x * p.x % MOD);
        return *this;
    }

    Arbitrary_Modint &operator/=(const Arbitrary_Modint &p) {
        *this *= p.inverse();
        return *this;
    }

    Arbitrary_Modint &operator%=(const Arbitrary_Modint &p) {
        assert(p.x == 0);
        return *this;
    }

    Arbitrary_Modint operator-() const {
        return Arbitrary_Modint(-x);
    }

    Arbitrary_Modint &operator++() {
        x++;
        if (x == MOD) x = 0;
        return *this;
    }

    Arbitrary_Modint &operator--() {
        if (x == 0) x = MOD;
        x--;
        return *this;
    }

    Arbitrary_Modint operator++(int) {
        Arbitrary_Modint result = *this;
        ++*this;
        return result;
    }

    Arbitrary_Modint operator--(int) {
        Arbitrary_Modint result = *this;
        --*this;
        return result;
    }

    friend Arbitrary_Modint operator+(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) {
        return Arbitrary_Modint(lhs) += rhs;
    }

    friend Arbitrary_Modint operator-(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) {
        return Arbitrary_Modint(lhs) -= rhs;
    }

    friend Arbitrary_Modint operator*(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) {
        return Arbitrary_Modint(lhs) *= rhs;
    }

    friend Arbitrary_Modint operator/(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) {
        return Arbitrary_Modint(lhs) /= rhs;
    }

    friend Arbitrary_Modint operator%(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) {
        assert(rhs.x == 0);
        return Arbitrary_Modint(lhs);
    }

    bool operator==(const Arbitrary_Modint &p) const {
        return x == p.x;
    }

    bool operator!=(const Arbitrary_Modint &p) const {
        return x != p.x;
    }

    bool operator<(const Arbitrary_Modint &rhs) {
        return x < rhs.x;
    }

    bool operator<=(const Arbitrary_Modint &rhs) {
        return x <= rhs.x;
    }

    bool operator>(const Arbitrary_Modint &rhs) {
        return x > rhs.x;
    }

    bool operator>=(const Arbitrary_Modint &rhs) {
        return x >= rhs.x;
    }

    Arbitrary_Modint inverse() const {
        int a = x, b = MOD, u = 1, v = 0, t;
        while (b > 0) {
            t = a / b;
            a -= t * b;
            u -= t * v;
            std::swap(a, b);
            std::swap(u, v);
        }
        return Arbitrary_Modint(u);
    }

    Arbitrary_Modint pow(int64_t k) const {
        Arbitrary_Modint ret(1);
        Arbitrary_Modint y(x);
        while (k > 0) {
            if (k & 1) ret *= y;
            y *= y;
            k >>= 1;
        }
        return ret;
    }

    friend std::ostream &operator<<(std::ostream &os, const Arbitrary_Modint &p) {
        return os << p.x;
    }

    friend std::istream &operator>>(std::istream &is, Arbitrary_Modint &p) {
        int64_t y;
        is >> y;
        p = Arbitrary_Modint(y);
        return (is);
    }

    static int get_mod() {
        return MOD;
    }
};
int Arbitrary_Modint::MOD = 998244353;

using modint9 = Modint<998244353>;
using modint1 = Modint<1000000007>;
using modint  = Arbitrary_Modint;
using mint    = modint9;

template <typename T>
T modinv(T a, T MOD) {
    T b = MOD;
    T u = 1;
    T v = 0;
    while (b > 0) {
        T t = a / b;
        a -= t * b;
        u -= t * v;
        std::swap(a, b);
        std::swap(u, v);
    }
    if (a != 1) return -1;
    if (u < 0) u += MOD;
    return u;
}

template <typename T>
std::vector<T> berlekampMessy(const std::vector<T> &A) {
    int n = A.size();
    std::vector<T> B(1, -1);
    std::vector<T> C(1, -1);
    T y = 1;
    for (int j = 1; j <= n; j++) {
        int l = C.size();
        int m = B.size();
        T x   = 0;
        for (int i = 0; i < l; i++) {
            x += C[i] * A[j - l + i];
        }
        B.push_back(0);
        m++;
        if (x == 0) continue;
        T freq = x / y;
        if (l < m) {
            std::vector<T> D(m - l, T(0));
            D.insert(D.end(), C.begin(), C.end());
            for (int i = 0; i < m; i++) {
                D[m - 1 - i] -= freq * B[m - 1 - i];
            }
            std::swap(B, C);
            std::swap(C, D);
            y = x;
        } else {
            for (int i = 0; i < m; i++) {
                C[l - 1 - i] -= freq * B[m - 1 - i];
            }
        }
    }

    std::reverse(C.begin(), C.end());
    for (auto &c : C) c = -c;
    return C;
}

/*
return x^n mod f
引数: f_reversed: f(x)の係数(逆順, f_reversed[0] = 1)
*/
template <typename T>
std::vector<T> monomial_mod_polynomial(long long n, const std::vector<T> &f_reversed) {
    assert(!f_reversed.empty() and f_reversed[0] == 1);
    int K = f_reversed.size() - 1;
    if (!K) return {};
    int D = 64 - __builtin_clzll(n);
    std::vector<T> ret(K, 0);
    ret[0]         = 1;
    auto self_conv = [](std::vector<T> &x) -> std::vector<T> {
        int d = x.size();
        std::vector<T> ret(2 * d - 1);
        for (int i = 0; i < d; i++) {
            ret[2 * i] += x[i] * x[i];
            for (int j = 0; j < i; j++) ret[i + j] += 2 * x[i] * x[j];
        }
        return ret;
    };
    for (int d = D - 1; d >= 0; d--) {
        ret = self_conv(ret);
        for (int i = 2 * K - 2; i >= K; i--) {
            for (int j = 1; j <= K; j++) {
                ret[i - j] -= ret[i] * f_reversed[j];
            }
        }
        ret.resize(K);
        if ((n >> d) & 1) {
            std::vector<T> c(K);
            c[0] = -ret[K - 1] * f_reversed[K];
            for (int i = 1; i < K; i++) {
                c[i] = ret[i - 1] - ret[K - 1] * f_reversed[K - i];
            }
            ret = c;
        }
    }
    return ret;
}

template <typename T>
T guess_kth_term(const std::vector<T> &A, long long k, bool debug = false) {
    assert(k >= 0);
    if (k < int(A.size())) return A[k];

    const auto F = berlekampMessy(A);
    if (debug) {
        std::cerr << "F.size() = " << F.size() << "\n" << std::flush;
    }
    const auto G = monomial_mod_polynomial<T>(k, F);
    T ret        = 0;
    for (size_t i = 0; i < G.size(); i++) ret += G[i] * A[i];
    return ret;
}
using mint2 = Modint<998244352>;

template <typename type>
struct Matrix {
    int n, m;
    std::vector<std::vector<type>> A;
    Matrix() = default;
    Matrix(int n, int m) : n(n), m(m), A(n, std::vector<type>(m, 0)) {}
    Matrix(int n) : n(n), m(n), A(n, std::vector<type>(n, 0)) {}
    Matrix(std::vector<std::vector<type>> A) : n(A.size()), m(A[0].size()), A(A) {}

    inline const std::vector<type> &operator[](int k) const {
        return (A.at(k));
    }

    inline std::vector<type> &operator[](int k) {
        return (A.at(k));
    }

    Matrix T() {
        Matrix<type> B(m, n);
        for (int i = 0; i < m; i++)
            for (int j = 0; j < n; j++) {
                B.A[i][j] = A[j][i];
            }
        return B;
    }

    Matrix &operator=(const std::vector<std::vector<type>> &B) {
        n = B.size();
        m = B[0].size();
        A = B;
        return *this;
    }

    Matrix &operator+=(const Matrix &B) {
        assert(n == int(B.A.size()));
        assert(m == int(B.A[0].size()));
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++) {
                this->A[i][j] += B[i][j];
            }
        return *this;
    }

    Matrix &operator-=(const Matrix &B) {
        assert(n == int(B.A.size()));
        assert(m == int(B.A[0].size()));
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++) {
                this->A[i][j] -= B[i][j];
            }
        return *this;
    }

    Matrix &operator*=(const Matrix &B) {
        int k = B[0].size();
        assert(m == int(B.A.size()));
        std::vector<std::vector<type>> C(n, std::vector<type>(k, 0));
        for (int i = 0; i < n; i++)
            for (int j = 0; j < k; j++) {
                for (int l = 0; l < m; l++) {
                    C[i][j] += this->A[i][l] * B[l][j];
                }
            }
        std::swap(this->A, C);
        this->m = k;
        return *this;
    }

    std::vector<type> operator*(const std::vector<type> &x) {
        assert(m == int(x.size()));
        std::vector<type> ret(n, 0);
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++) ret[i] += this->A[i][j] * x[j];
        return ret;
    }

    template <typename Ti>
    Matrix &operator*=(const Ti x) {
        for (auto &row : A) {
            for (auto &e : row) {
                e *= x;
            }
        }
        return *this;
    }

    Matrix operator-() {
        return (Matrix(*this) *= -1);
    }

    Matrix operator+(const Matrix &B) const {
        return (Matrix(*this) += B);
    }

    Matrix operator-(const Matrix &B) const {
        return (Matrix(*this) -= B);
    }

    Matrix operator*(const Matrix &B) const {
        return (Matrix(*this) *= B);
    }

    type det() {
        auto arr = A;
        assert(n == m);
        type ret = 1;
        for (int i = 0; i < n; i++) {
            if (arr[i][i] == 0) {
                bool ng = true;
                for (int j = i + 1; j < n; j++) {
                    if (arr[j][i] == 0) continue;
                    swap(arr[i], arr[j]);
                    ret *= -1;
                    ng = false;
                    break;
                }
                if (ng) return 0;
            }
            ret *= arr[i][i];
            type inv = type(1) / arr[i][i];
            for (int j = i; j < n; j++) arr[i][j] *= inv;
            for (int j = i + 1; j < n; j++) {
                type x = arr[j][i];
                for (int k = i; k < n; k++) {
                    arr[j][k] -= arr[i][k] * x;
                }
            }
        }
        return ret;
    }

    void I() {
        assert(n == m);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                if (i == j)
                    A[i][j] = 1;
                else
                    A[i][j] = 0;
            }
        }
    }

    Matrix<type> inv() {
        assert(n == m);
        Matrix<type> ret(n);
        ret.I();
        auto &B  = ret.A;
        auto arr = A;
        for (int j = 0; j < n; j++) {
            int ii = -1;
            for (int i = j; i < n; i++) {
                if (arr[i][j] != 0) {
                    ii = i;
                    break;
                }
            }
            if (ii == -1) {
                return {};
            }
            swap(arr[j], arr[ii]);
            swap(B[j], B[ii]);
            ii       = j;
            type inv = type(1) / arr[ii][j];

            for (int jj = 0; jj < n; jj++) {
                B[ii][jj] *= inv;
                arr[ii][jj] *= inv;
            }

            for (int i = 0; i < n; i++) {
                if (i == ii) continue;
                type t = arr[i][j];
                for (int jj = 0; jj < n; jj++) {
                    arr[i][jj] -= arr[ii][jj] * t;
                    B[i][jj] -= B[ii][jj] * t;
                }
            }
        }
        return ret;
    }

    int choose_pivot(int h, int c) const {
        for (int j = h; j < n; j++) {
            if (A[j][c] != type(0)) return j;
        }
        return -1;
    }

    int rank() const {
        auto arr = *this;
        if (arr.n < arr.m) {
            arr = arr.T();
        }
        int ret = 0;
        for (int i = 0; i < arr.m; i++) {
            int j = arr.choose_pivot(ret, i);
            if (j == -1) continue;
            swap(arr[ret], arr[j]);
            type inv = type(1) / arr[ret][i];
            for (int k = i; k < arr.m; k++) {
                arr[ret][k] *= inv;
            }
            for (int j = ret + 1; j < arr.n; j++) {
                type x = arr[j][i];
                for (int k = i; k < arr.m; k++) {
                    arr[j][k] -= arr[ret][k] * x;
                }
            }
            ret++;
        }
        return ret;
    }

    Matrix<type> pow(long long k) {
        assert(n == m);
        Matrix<type> B(n);
        B.I();
        Matrix<type> A(*this);
        while (k) {
            if (k & 1) B *= A;
            A *= A;
            k >>= 1;
        }
        return B;
    }

    friend std::ostream &operator<<(std::ostream &os, const Matrix &p) {
        for (int i = 0; i < p.n; i++) {
            for (auto &x : p.A[i]) {
                os << x << " ";
            }
            if (i != p.n - 1) {
                os << "\n";
            }
        }
        return (os);
    }

    friend std::istream &operator>>(std::istream &is, Matrix &p) {
        for (auto &row : p.A) {
            for (auto &x : row) {
                is >> x;
            }
        }
        return (is);
    }
};

template <typename T>
Matrix<T> Matrix_exp(Matrix<T> A, Matrix<T> B, long long k) {
    assert(A.A.size() == A[0].size());
    assert(A.A.size() == B.A.size());
    assert(B.A[0].size() == 1u);
    while (k > 0) {
        if (k & 1) B = A * B;
        A *= A;
        k >>= 1;
    }
    return B;
}

template <typename T>
std::vector<T> Matrix_exp(Matrix<T> A, std::vector<T> B, long long k) {
    assert(A.A.size() == A[0].size());
    assert(A.A.size() == B.size());
    while (k > 0) {
        if (k & 1) {
            B = A * B;
        }
        A *= A;
        k >>= 1;
    }
    return B;
}

void solve() {
    LL(n);

    if (n <= 1000) {
        mint now   = 1;
        int length = 1;
        fori(i, 2, n + 1) {
            auto x = i;
            int le = 0;
            while (x) {
                x /= 10;
                le++;
            }

            mint nex = now;
            nex += i * mint(10).pow(length);
            nex += now * mint(10).pow(length + le);
            length = 2 * length + le;
            length %= 998244352;

            now = nex;
        }

        print(now);
        return;
    }

    mint now   = 1;
    int length = 1;
    fori(i, 2, 1000) {
        auto x = i;
        int le = 0;
        while (x) {
            x /= 10;
            le++;
        }

        mint nex = now;
        nex += i * mint(10).pow(length);
        nex += now * mint(10).pow(length + le);
        length = 2 * length + le;
        length %= 998244352;

        now = nex;
    }

    ll x = 1000;
    n++;
    int le = 4;
    while (x < n) {
        ll y = x * 10;
        chmin(y, n);

        vec(mint, A, 500);
        mint2 bef_length = length;
        fori(i, x, x + 500) {
            mint nex = now;
            nex += i * mint(10).pow(length);
            nex += now * mint(10).pow(length + le);
            length = 2 * length + le;
            length %= 998244352;

            now      = nex;
            A[i - x] = now;
        }

        now = guess_kth_term(A, y - x - 1, true);

        Matrix<mint2> B({
            {2, le},
            {0, 1},
        });
        B      = B.pow(y - x);
        length = (B[0][0] * bef_length + B[0][1]).x;

        le++;
        x = y;
    }
    print(now);
}

int main() {
#ifndef INTERACTIVE
    std::cin.tie(0)->sync_with_stdio(0);
#endif
    // std::cout << std::fixed << std::setprecision(12);
    int t;
    t = 1;
    // std::cin >> t;
    while (t--) solve();
    return 0;
}

// // #pragma GCC target("avx2")
// // #pragma GCC optimize("O3")
// // #pragma GCC optimize("unroll-loops")
// // #define INTERACTIVE
//
// #include "kyopro-cpp/template.hpp"
//
// #include "misc/Modint.hpp"
// using mint = modint9;
// #include "polynomial/guess_kth_term.hpp"
// using mint2 = Modint<998244352>;
// #include "matrix/Matrix_exp.hpp"
//
// void solve() {
//     LL(n);
//
//     if (n <= 1000) {
//         mint now   = 1;
//         int length = 1;
//         fori(i, 2, n + 1) {
//             auto x = i;
//             int le = 0;
//             while (x) {
//                 x /= 10;
//                 le++;
//             }
//
//             mint nex = now;
//             nex += i * mint(10).pow(length);
//             nex += now * mint(10).pow(length + le);
//             length = 2 * length + le;
//             length %= 998244352;
//
//             now = nex;
//         }
//
//         print(now);
//         return;
//     }
//
//     mint now   = 1;
//     int length = 1;
//     fori(i, 2, 1000) {
//         auto x = i;
//         int le = 0;
//         while (x) {
//             x /= 10;
//             le++;
//         }
//
//         mint nex = now;
//         nex += i * mint(10).pow(length);
//         nex += now * mint(10).pow(length + le);
//         length = 2 * length + le;
//         length %= 998244352;
//
//         now = nex;
//     }
//
//     ll x = 1000;
//     n++;
//     int le = 4;
//     while (x < n) {
//         ll y = x * 10;
//         chmin(y, n);
//
//         vec(mint, A, 500);
//         mint2 bef_length = length;
//         fori(i, x, x + 500) {
//             mint nex = now;
//             nex += i * mint(10).pow(length);
//             nex += now * mint(10).pow(length + le);
//             length = 2 * length + le;
//             length %= 998244352;
//
//             now      = nex;
//             A[i - x] = now;
//         }
//
//         now = guess_kth_term(A, y - x - 1, true);
//
//         Matrix<mint2> B({
//             {2, le},
//             {0, 1},
//         });
//         B      = B.pow(y - x);
//         length = (B[0][0] * bef_length + B[0][1]).x;
//
//         le++;
//         x = y;
//     }
//     print(now);
// }
//
// int main() {
// #ifndef INTERACTIVE
//     std::cin.tie(0)->sync_with_stdio(0);
// #endif
//     // std::cout << std::fixed << std::setprecision(12);
//     int t;
//     t = 1;
//     // std::cin >> t;
//     while (t--) solve();
//     return 0;
// }