#include "testlib.h"
#line 2 "cpplib/util/template.hpp"
#ifdef LOCAL
#define _GLIBCXX_DEBUG
#endif
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")
#pragma GCC target("avx2")
#include <bits/stdc++.h>
using namespace std;
#line 1 "cpplib/util/ioutil.hpp"
// template <class Head,class... Args>
// std::ostream& output(std::ostream& out,const Head& head,const Args&... args){
//     out>>head;
//     return output(head,args...);
// }
// template <class Head>
// std::ostream& output(std::ostream& out,const Head& head){
//     out>>head;
//     return out;
// }

template <typename T, typename E>
std::ostream &operator<<(std::ostream &out, std::pair<T, E> v) {
    out << "(" << v.first << "," << v.second << ")";
    return out;
}

// template <class... Args>
// ostream& operator<<(ostream& out,std::tuple<Args...>v){
//     std::apply(output,v);
//     return out;
// }
#line 11 "cpplib/util/template.hpp"
struct __INIT__ {
    __INIT__() {
        cin.tie(0);
        ios::sync_with_stdio(false);
        cout << fixed << setprecision(15);
    }
} __INIT__;
typedef long long lint;
constexpr long long INF = 1LL << 60;
constexpr int IINF = 1 << 30;
constexpr double EPS = 1e-10;
#ifndef REACTIVE
#define endl '\n';
#endif
typedef vector<lint> vec;
typedef vector<vector<lint>> mat;
typedef vector<vector<vector<lint>>> mat3;
typedef vector<string> svec;
typedef vector<vector<string>> smat;
template <typename T> using V = vector<T>;
template <typename T> using VV = V<V<T>>;
#define output(t)                                                              \
    {                                                                          \
        bool f = 0;                                                            \
        for (auto val : (t)) {                                                 \
            cout << (f ? " " : "") << val;                                     \
            f = 1;                                                             \
        }                                                                      \
        cout << endl;                                                          \
    }
#define output2(t)                                                             \
    {                                                                          \
        for (auto i : t)                                                       \
            output(i);                                                         \
    }
#define debug(t)                                                               \
    {                                                                          \
        bool f = 0;                                                            \
        for (auto i : t) {                                                     \
            cerr << (f ? " " : "") << i;                                       \
            f = 1;                                                             \
        }                                                                      \
        cerr << endl;                                                          \
    }
#define debug2(t)                                                              \
    {                                                                          \
        for (auto i : t)                                                       \
            debug(i);                                                          \
    }
#define loop(n) for (long long _ = 0; _ < (long long)(n); ++_)
#define _overload4(_1, _2, _3, _4, name, ...) name
#define __rep(i, a) repi(i, 0, a, 1)
#define _rep(i, a, b) repi(i, a, b, 1)
#define repi(i, a, b, c)                                                       \
    for (long long i = (long long)(a); i < (long long)(b); i += c)
#define rep(...) _overload4(__VA_ARGS__, repi, _rep, __rep)(__VA_ARGS__)
#define _overload3_rev(_1, _2, _3, name, ...) name
#define _rep_rev(i, a) repi_rev(i, 0, a)
#define repi_rev(i, a, b)                                                      \
    for (long long i = (long long)(b)-1; i >= (long long)(a); --i)
#define rrep(...) _overload3_rev(__VA_ARGS__, repi_rev, _rep_rev)(__VA_ARGS__)

#define all(n) begin(n), end(n)
template <typename T, typename E> bool chmin(T &s, const E &t) {
    bool res = s > t;
    s = min<T>(s, t);
    return res;
}
template <typename T, typename E> bool chmax(T &s, const E &t) {
    bool res = s < t;
    s = max<T>(s, t);
    return res;
}
const vector<lint> dx = {1, 0, -1, 0, 1, 1, -1, -1};
const vector<lint> dy = {0, 1, 0, -1, 1, -1, 1, -1};
#define SUM(v) accumulate(all(v), 0LL)
#if __cplusplus >= 201703L
template <typename T, typename... Args>
auto make_vector(T x, int arg, Args... args) {
    if constexpr (sizeof...(args) == 0)
        return vector<T>(arg, x);
    else
        return vector(arg, make_vector<T>(x, args...));
}
#endif
#define bit(n, a) ((n >> a) & 1)
#define extrep(v, ...) for (auto v : make_mat_impl({__VA_ARGS__}))
vector<vector<long long>> make_mat_impl(vector<long long> v) {
    if (v.empty())
        return vector<vector<long long>>(1, vector<long long>());
    long long n = v.back();
    v.pop_back();
    vector<vector<long long>> ret;
    vector<vector<long long>> tmp = make_mat_impl(v);
    for (auto e : tmp)
        for (long long i = 0; i < n; ++i) {
            ret.push_back(e);
            ret.back().push_back(i);
        }
    return ret;
}
using graph = vector<vector<int>>;
template <typename T> using graph_w = vector<vector<pair<int, T>>>;

#if __cplusplus >= 201703L
constexpr inline long long powll(long long a, long long b) {
    long long res = 1;
    while (b--)
        res *= a;
    return res;
}
#endif

template <typename T, typename E>
pair<T, E> &operator+=(pair<T, E> &s, const pair<T, E> &t) {
    s.first += t.first;
    s.second += t.second;
    return s;
}
template <typename T, typename E>
pair<T, E> &operator-=(pair<T, E> &s, const pair<T, E> &t) {
    s.first -= t.first;
    s.second -= t.second;
    return s;
}
template <typename T, typename E>
pair<T, E> operator+(const pair<T, E> &s, const pair<T, E> &t) {
    auto res = s;
    return res += t;
}
template <typename T, typename E>
pair<T, E> operator-(const pair<T, E> &s, const pair<T, E> &t) {
    auto res = s;
    return res -= t;
}
#define BEGIN_STACK_EXTEND(size)                                               \
    void *stack_extend_memory_ = malloc(size);                                 \
    void *stack_extend_origin_memory_;                                         \
    char *stack_extend_dummy_memory_ = (char *)alloca(                         \
        (1 + (int)(((long long)stack_extend_memory_) & 127)) * 16);            \
    *stack_extend_dummy_memory_ = 0;                                           \
    asm volatile("mov %%rsp, %%rbx\nmov %%rax, %%rsp"                          \
                 : "=b"(stack_extend_origin_memory_)                           \
                 : "a"((char *)stack_extend_memory_ + (size)-1024));
#define END_STACK_EXTEND                                                       \
    asm volatile("mov %%rax, %%rsp" ::"a"(stack_extend_origin_memory_));       \
    free(stack_extend_memory_);
int floor_pow(int n) { return n ? 31 - __builtin_clz(n) : 0; }
#line 3 "main.cpp"

#include <algorithm>
#include <array>
#include <cassert>
#include <type_traits>
#include <vector>

#ifdef _MSC_VER
#include <intrin.h>
#endif

#if __cplusplus >= 202002L
#include <bit>
#endif

namespace atcoder {

namespace internal {

#if __cplusplus >= 202002L

using std::bit_ceil;

#else

unsigned int bit_ceil(unsigned int n) {
    unsigned int x = 1;
    while (x < (unsigned int)(n))
        x *= 2;
    return x;
}

#endif

int countr_zero(unsigned int n) {
#ifdef _MSC_VER
    unsigned long index;
    _BitScanForward(&index, n);
    return index;
#else
    return __builtin_ctz(n);
#endif
}

constexpr int countr_zero_constexpr(unsigned int n) {
    int x = 0;
    while (!(n & (1 << x)))
        x++;
    return x;
}

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

#include <utility>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0)
        x += m;
    return x;
}

struct barrett {
    unsigned int _m;
    unsigned long long im;

    explicit barrett(unsigned int m)
        : _m(m), im((unsigned long long)(-1) / m + 1) {}

    unsigned int umod() const { return _m; }

    unsigned int mul(unsigned int a, unsigned int b) const {

        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
            (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned long long y = x * _m;
        return (unsigned int)(z - y + (z < y ? _m : 0));
    }
};

constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
    if (m == 1)
        return 0;
    unsigned int _m = (unsigned int)(m);
    unsigned long long r = 1;
    unsigned long long y = safe_mod(x, m);
    while (n) {
        if (n & 1)
            r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

constexpr bool is_prime_constexpr(int n) {
    if (n <= 1)
        return false;
    if (n == 2 || n == 7 || n == 61)
        return true;
    if (n % 2 == 0)
        return false;
    long long d = n - 1;
    while (d % 2 == 0)
        d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
    a = safe_mod(a, b);
    if (a == 0)
        return {b, 0};

    long long s = b, t = a;
    long long m0 = 0, m1 = 1;

    while (t) {
        long long u = s / t;
        s -= t * u;
        m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b

        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    if (m0 < 0)
        m0 += b / s;
    return {s, m0};
}

constexpr int primitive_root_constexpr(int m) {
    if (m == 2)
        return 1;
    if (m == 167772161)
        return 3;
    if (m == 469762049)
        return 3;
    if (m == 754974721)
        return 11;
    if (m == 998244353)
        return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while (x % 2 == 0)
        x /= 2;
    for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
            divs[cnt++] = i;
            while (x % i == 0) {
                x /= i;
            }
        }
    }
    if (x > 1) {
        divs[cnt++] = x;
    }
    for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
            if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if (ok)
            return g;
    }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

unsigned long long floor_sum_unsigned(unsigned long long n,
                                      unsigned long long m,
                                      unsigned long long a,
                                      unsigned long long b) {
    unsigned long long ans = 0;
    while (true) {
        if (a >= m) {
            ans += n * (n - 1) / 2 * (a / m);
            a %= m;
        }
        if (b >= m) {
            ans += n * (b / m);
            b %= m;
        }

        unsigned long long y_max = a * n + b;
        if (y_max < m)
            break;
        n = (unsigned long long)(y_max / m);
        b = (unsigned long long)(y_max % m);
        std::swap(m, a);
    }
    return ans;
}

} // namespace internal

} // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

namespace atcoder {

namespace internal {

#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value ||
                                  std::is_same<T, __int128>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value ||
                                  std::is_same<T, unsigned __int128>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value, __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral =
    typename std::conditional<std::is_integral<T>::value ||
                                  is_signed_int128<T>::value ||
                                  is_unsigned_int128<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_signed_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_signed<T>::value) ||
                                  is_signed_int128<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value, make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value, std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

#else

template <class T> using is_integral = typename std::is_integral<T>;

template <class T>
using is_signed_int =
    typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<is_integral<T>::value &&
                                  std::is_unsigned<T>::value,
                              std::true_type, std::false_type>::type;

template <class T>
using to_unsigned =
    typename std::conditional<is_signed_int<T>::value, std::make_unsigned<T>,
                              std::common_type<T>>::type;

#endif

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

} // namespace internal

} // namespace atcoder

namespace atcoder {

namespace internal {

struct modint_base {};
struct static_modint_base : modint_base {};

template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

} // namespace internal

template <int m, std::enable_if_t<(1 <= m)> * = nullptr>
struct static_modint : internal::static_modint_base {
    using mint = static_modint;

  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    static_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T> * = nullptr>
    static_modint(T v) {
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0)
            x += umod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T> * = nullptr>
    static_modint(T v) {
        _v = (unsigned int)(v % umod());
    }

    unsigned int val() const { return _v; }

    mint &operator++() {
        _v++;
        if (_v == umod())
            _v = 0;
        return *this;
    }
    mint &operator--() {
        if (_v == 0)
            _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint &operator+=(const mint &rhs) {
        _v += rhs._v;
        if (_v >= umod())
            _v -= umod();
        return *this;
    }
    mint &operator-=(const mint &rhs) {
        _v -= rhs._v;
        if (_v >= umod())
            _v += umod();
        return *this;
    }
    mint &operator*=(const mint &rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        _v = (unsigned int)(z % umod());
        return *this;
    }
    mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1)
                r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = internal::inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }

    friend mint operator+(const mint &lhs, const mint &rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint &lhs, const mint &rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint &lhs, const mint &rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint &lhs, const mint &rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint &lhs, const mint &rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint &lhs, const mint &rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = internal::is_prime<m>;
};

template <int id> struct dynamic_modint : internal::modint_base {
    using mint = dynamic_modint;

  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = internal::barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T> * = nullptr>
    dynamic_modint(T v) {
        long long x = (long long)(v % (long long)(mod()));
        if (x < 0)
            x += mod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T> * = nullptr>
    dynamic_modint(T v) {
        _v = (unsigned int)(v % mod());
    }

    unsigned int val() const { return _v; }

    mint &operator++() {
        _v++;
        if (_v == umod())
            _v = 0;
        return *this;
    }
    mint &operator--() {
        if (_v == 0)
            _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint &operator+=(const mint &rhs) {
        _v += rhs._v;
        if (_v >= umod())
            _v -= umod();
        return *this;
    }
    mint &operator-=(const mint &rhs) {
        _v += mod() - rhs._v;
        if (_v >= umod())
            _v -= umod();
        return *this;
    }
    mint &operator*=(const mint &rhs) {
        _v = bt.mul(_v, rhs._v);
        return *this;
    }
    mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1)
                r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        auto eg = internal::inv_gcd(_v, mod());
        assert(eg.first == 1);
        return eg.second;
    }

    friend mint operator+(const mint &lhs, const mint &rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint &lhs, const mint &rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint &lhs, const mint &rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint &lhs, const mint &rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint &lhs, const mint &rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint &lhs, const mint &rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static internal::barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt(998244353);

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

namespace internal {

template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

} // namespace internal

} // namespace atcoder

namespace atcoder {

namespace internal {

template <class mint, int g = internal::primitive_root<mint::mod()>,
          internal::is_static_modint_t<mint> * = nullptr>
struct fft_info {
    static constexpr int rank2 = countr_zero_constexpr(mint::mod() - 1);
    std::array<mint, rank2 + 1> root;  // root[i]^(2^i) == 1
    std::array<mint, rank2 + 1> iroot; // root[i] * iroot[i] == 1

    std::array<mint, std::max(0, rank2 - 2 + 1)> rate2;
    std::array<mint, std::max(0, rank2 - 2 + 1)> irate2;

    std::array<mint, std::max(0, rank2 - 3 + 1)> rate3;
    std::array<mint, std::max(0, rank2 - 3 + 1)> irate3;

    fft_info() {
        root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
        iroot[rank2] = root[rank2].inv();
        for (int i = rank2 - 1; i >= 0; i--) {
            root[i] = root[i + 1] * root[i + 1];
            iroot[i] = iroot[i + 1] * iroot[i + 1];
        }

        {
            mint prod = 1, iprod = 1;
            for (int i = 0; i <= rank2 - 2; i++) {
                rate2[i] = root[i + 2] * prod;
                irate2[i] = iroot[i + 2] * iprod;
                prod *= iroot[i + 2];
                iprod *= root[i + 2];
            }
        }
        {
            mint prod = 1, iprod = 1;
            for (int i = 0; i <= rank2 - 3; i++) {
                rate3[i] = root[i + 3] * prod;
                irate3[i] = iroot[i + 3] * iprod;
                prod *= iroot[i + 3];
                iprod *= root[i + 3];
            }
        }
    }
};

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
void butterfly(std::vector<mint> &a) {
    int n = int(a.size());
    int h = internal::countr_zero((unsigned int)n);

    static const fft_info<mint> info;

    int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len < h) {
        if (h - len == 1) {
            int p = 1 << (h - len - 1);
            mint rot = 1;
            for (int s = 0; s < (1 << len); s++) {
                int offset = s << (h - len);
                for (int i = 0; i < p; i++) {
                    auto l = a[i + offset];
                    auto r = a[i + offset + p] * rot;
                    a[i + offset] = l + r;
                    a[i + offset + p] = l - r;
                }
                if (s + 1 != (1 << len))
                    rot *= info.rate2[countr_zero(~(unsigned int)(s))];
            }
            len++;
        } else {
            int p = 1 << (h - len - 2);
            mint rot = 1, imag = info.root[2];
            for (int s = 0; s < (1 << len); s++) {
                mint rot2 = rot * rot;
                mint rot3 = rot2 * rot;
                int offset = s << (h - len);
                for (int i = 0; i < p; i++) {
                    auto mod2 = 1ULL * mint::mod() * mint::mod();
                    auto a0 = 1ULL * a[i + offset].val();
                    auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
                    auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
                    auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
                    auto a1na3imag =
                        1ULL * mint(a1 + mod2 - a3).val() * imag.val();
                    auto na2 = mod2 - a2;
                    a[i + offset] = a0 + a2 + a1 + a3;
                    a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
                    a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
                    a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
                }
                if (s + 1 != (1 << len))
                    rot *= info.rate3[countr_zero(~(unsigned int)(s))];
            }
            len += 2;
        }
    }
}

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
void butterfly_inv(std::vector<mint> &a) {
    int n = int(a.size());
    int h = internal::countr_zero((unsigned int)n);

    static const fft_info<mint> info;

    int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
    while (len) {
        if (len == 1) {
            int p = 1 << (h - len);
            mint irot = 1;
            for (int s = 0; s < (1 << (len - 1)); s++) {
                int offset = s << (h - len + 1);
                for (int i = 0; i < p; i++) {
                    auto l = a[i + offset];
                    auto r = a[i + offset + p];
                    a[i + offset] = l + r;
                    a[i + offset + p] =
                        (unsigned long long)(mint::mod() + l.val() - r.val()) *
                        irot.val();
                    ;
                }
                if (s + 1 != (1 << (len - 1)))
                    irot *= info.irate2[countr_zero(~(unsigned int)(s))];
            }
            len--;
        } else {
            int p = 1 << (h - len);
            mint irot = 1, iimag = info.iroot[2];
            for (int s = 0; s < (1 << (len - 2)); s++) {
                mint irot2 = irot * irot;
                mint irot3 = irot2 * irot;
                int offset = s << (h - len + 2);
                for (int i = 0; i < p; i++) {
                    auto a0 = 1ULL * a[i + offset + 0 * p].val();
                    auto a1 = 1ULL * a[i + offset + 1 * p].val();
                    auto a2 = 1ULL * a[i + offset + 2 * p].val();
                    auto a3 = 1ULL * a[i + offset + 3 * p].val();

                    auto a2na3iimag =
                        1ULL *
                        mint((mint::mod() + a2 - a3) * iimag.val()).val();

                    a[i + offset] = a0 + a1 + a2 + a3;
                    a[i + offset + 1 * p] =
                        (a0 + (mint::mod() - a1) + a2na3iimag) * irot.val();
                    a[i + offset + 2 * p] =
                        (a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) *
                        irot2.val();
                    a[i + offset + 3 * p] =
                        (a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) *
                        irot3.val();
                }
                if (s + 1 != (1 << (len - 2)))
                    irot *= info.irate3[countr_zero(~(unsigned int)(s))];
            }
            len -= 2;
        }
    }
}

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
std::vector<mint> convolution_naive(const std::vector<mint> &a,
                                    const std::vector<mint> &b) {
    int n = int(a.size()), m = int(b.size());
    std::vector<mint> ans(n + m - 1);
    if (n < m) {
        for (int j = 0; j < m; j++) {
            for (int i = 0; i < n; i++) {
                ans[i + j] += a[i] * b[j];
            }
        }
    } else {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                ans[i + j] += a[i] * b[j];
            }
        }
    }
    return ans;
}

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
std::vector<mint> convolution_fft(std::vector<mint> a, std::vector<mint> b) {
    int n = int(a.size()), m = int(b.size());
    int z = (int)internal::bit_ceil((unsigned int)(n + m - 1));
    a.resize(z);
    internal::butterfly(a);
    b.resize(z);
    internal::butterfly(b);
    for (int i = 0; i < z; i++) {
        a[i] *= b[i];
    }
    internal::butterfly_inv(a);
    a.resize(n + m - 1);
    mint iz = mint(z).inv();
    for (int i = 0; i < n + m - 1; i++)
        a[i] *= iz;
    return a;
}

} // namespace internal

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
std::vector<mint> convolution(std::vector<mint> &&a, std::vector<mint> &&b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m)
        return {};

    int z = (int)internal::bit_ceil((unsigned int)(n + m - 1));
    assert((mint::mod() - 1) % z == 0);

    if (std::min(n, m) <= 60)
        return convolution_naive(a, b);
    return internal::convolution_fft(a, b);
}
template <class mint, internal::is_static_modint_t<mint> * = nullptr>
std::vector<mint> convolution(const std::vector<mint> &a,
                              const std::vector<mint> &b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m)
        return {};

    int z = (int)internal::bit_ceil((unsigned int)(n + m - 1));
    assert((mint::mod() - 1) % z == 0);

    if (std::min(n, m) <= 60)
        return convolution_naive(a, b);
    return internal::convolution_fft(a, b);
}

template <unsigned int mod = 998244353, class T,
          std::enable_if_t<internal::is_integral<T>::value> * = nullptr>
std::vector<T> convolution(const std::vector<T> &a, const std::vector<T> &b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m)
        return {};

    using mint = static_modint<mod>;

    int z = (int)internal::bit_ceil((unsigned int)(n + m - 1));
    assert((mint::mod() - 1) % z == 0);

    std::vector<mint> a2(n), b2(m);
    for (int i = 0; i < n; i++) {
        a2[i] = mint(a[i]);
    }
    for (int i = 0; i < m; i++) {
        b2[i] = mint(b[i]);
    }
    auto c2 = convolution(std::move(a2), std::move(b2));
    std::vector<T> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
        c[i] = c2[i].val();
    }
    return c;
}

std::vector<long long> convolution_ll(const std::vector<long long> &a,
                                      const std::vector<long long> &b) {
    int n = int(a.size()), m = int(b.size());
    if (!n || !m)
        return {};

    static constexpr unsigned long long MOD1 = 754974721; // 2^24
    static constexpr unsigned long long MOD2 = 167772161; // 2^25
    static constexpr unsigned long long MOD3 = 469762049; // 2^26
    static constexpr unsigned long long M2M3 = MOD2 * MOD3;
    static constexpr unsigned long long M1M3 = MOD1 * MOD3;
    static constexpr unsigned long long M1M2 = MOD1 * MOD2;
    static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;

    static constexpr unsigned long long i1 =
        internal::inv_gcd(MOD2 * MOD3, MOD1).second;
    static constexpr unsigned long long i2 =
        internal::inv_gcd(MOD1 * MOD3, MOD2).second;
    static constexpr unsigned long long i3 =
        internal::inv_gcd(MOD1 * MOD2, MOD3).second;

    static constexpr int MAX_AB_BIT = 24;
    static_assert(MOD1 % (1ull << MAX_AB_BIT) == 1,
                  "MOD1 isn't enough to support an array length of 2^24.");
    static_assert(MOD2 % (1ull << MAX_AB_BIT) == 1,
                  "MOD2 isn't enough to support an array length of 2^24.");
    static_assert(MOD3 % (1ull << MAX_AB_BIT) == 1,
                  "MOD3 isn't enough to support an array length of 2^24.");
    assert(n + m - 1 <= (1 << MAX_AB_BIT));

    auto c1 = convolution<MOD1>(a, b);
    auto c2 = convolution<MOD2>(a, b);
    auto c3 = convolution<MOD3>(a, b);

    std::vector<long long> c(n + m - 1);
    for (int i = 0; i < n + m - 1; i++) {
        unsigned long long x = 0;
        x += (c1[i] * i1) % MOD1 * M2M3;
        x += (c2[i] * i2) % MOD2 * M1M3;
        x += (c3[i] * i3) % MOD3 * M1M2;
        long long diff =
            c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
        if (diff < 0)
            diff += MOD1;
        static constexpr unsigned long long offset[5] = {
            0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
        x -= offset[diff % 5];
        c[i] = x;
    }

    return c;
}

} // namespace atcoder

#line 2 "cpplib/math/ACL_modint998244353.hpp"
using mint = atcoder::modint998244353;
#line 4 "cpplib/math/ACL_modint_base.hpp"

std::ostream &operator<<(std::ostream &lhs, const mint &rhs) noexcept {
    lhs << rhs.val();
    return lhs;
}
std::istream &operator>>(std::istream &lhs, mint &rhs) noexcept {
    long long x;
    lhs >> x;
    rhs = x;
    return lhs;
}

int MOD_NOW = -1;
int FACT_TABLE_SIZE = 0;
std::vector<mint> fact_table, fact_inv_table;

void update(int x) {
    if (MOD_NOW != mint::mod() || FACT_TABLE_SIZE == 0) {
        fact_table.assign(1, 1);
        fact_inv_table.assign(1, 1);
        FACT_TABLE_SIZE = 1;
        MOD_NOW = mint::mod();
    }
    while (FACT_TABLE_SIZE <= x) {
        fact_table.resize(FACT_TABLE_SIZE * 2);
        fact_inv_table.resize(FACT_TABLE_SIZE * 2);
        for (int i = FACT_TABLE_SIZE; i < FACT_TABLE_SIZE * 2; ++i) {
            fact_table[i] = fact_table[i - 1] * i;
        }
        fact_inv_table[FACT_TABLE_SIZE * 2 - 1] =
            fact_table[FACT_TABLE_SIZE * 2 - 1].inv();
        for (int i = FACT_TABLE_SIZE * 2 - 2; i >= FACT_TABLE_SIZE; --i) {
            fact_inv_table[i] = fact_inv_table[i + 1] * (i + 1);
        }
        FACT_TABLE_SIZE *= 2;
    }
}

inline mint fact(int x) {
    assert(x >= 0);
    update(x);
    return fact_table[x];
}
inline mint fact_inv(int x) {
    assert(x >= 0);
    update(x);
    return fact_inv_table[x];
}
inline mint comb(int x, int y) {
    if (x < 0 || x < y || y < 0)
        return 0;
    return fact(x) * fact_inv(y) * fact_inv(x - y);
}
inline mint perm(int x, int y) { return fact(x) * fact_inv(x - y); }

// x個のグループにy個のものを分ける場合の数
inline mint multi_comb(int x, int y) {
    if (y == 0 && x >= 0)
        return 1;
    if (y < 0 || x <= 0)
        return 0;
    return comb(x + y - 1, y);
}
#line 2 "cpplib/math/poly.hpp"

using poly = vector<mint>;
int size(const poly &x) { return x.size(); }
poly shrink(poly x) {
    while (size(x) >= 1 && x.back().val() == 0)
        x.pop_back();
    return x;
}

poly pre(const poly &x, int n) {
    auto res = x;
    res.resize(n);
    return res;
}
poly operator+(const poly &x, const poly &y) {
    poly res(max(x.size(), y.size()));
    rep(i, 0, x.size()) res[i] += x[i];
    rep(i, 0, y.size()) res[i] += y[i];
    return res;
}
poly &operator*=(poly &x, const mint &y) {
    rep(i, 0, x.size()) x[i] *= y;
    return x;
}
poly operator*(poly x, const mint &y) { return x *= y; }

poly operator-(const poly &x) {
    poly res(x.size());
    rep(i, 0, x.size()) res[i] = -x[i];
    return res;
}
poly operator-(const poly &x, const poly &y) { return x + (-y); }

poly operator*(const poly &x, const poly &y) {
    // if(x.size()+y.size()<=20){
    //     poly res(x.size()+y.size()-1);
    //     rep(i,x.size())rep(j,y.size()){
    //         res[i+j]+=x[i]*y[j];
    //     }
    //     return res;
    // }
    return atcoder::convolution(x, y);
}

// poly operator*(const poly&x,const poly&y){
//     poly res(x.size()+y.size()-1);
//     for(int i=0;i<x.size();++i){
//         for(int j=0;j<y.size();++j){
//             res[i+j]+=x[i]*y[j];
//         }
//     }
//     return res;
// }

// poly operator*(const poly&x,const poly&y){
//     return convolution(x,y);
// }

poly &operator+=(poly &x, const poly &y) { return x = (x + y); }
poly &operator-=(poly &x, const poly &y) { return x = (x - y); }
poly &operator*=(poly &x, const poly &y) { return x = (x * y); }
istream &operator>>(istream &in, poly &y) {
    int n = size(y);
    rep(i, 0, n) in >> y[i];
    return in;
}
ostream &operator<<(ostream &out, const poly &y) {
    int n = size(y);
    rep(i, 0, n) {
        if (i)
            out << ' ';
        out << y[i].val();
    }
    return out;
}
poly diff(const poly &x) {
    int n = size(x);
    poly res(n - 1);
    rep(i, 0, n - 1) res[i] = x[i + 1] * (i + 1);
    return res;
}
poly integrate(const poly &x) {
    int n = size(x);
    poly res(n + 1);
    rep(i, 1, n + 1) res[i] = x[i - 1] / i;
    return res;
}

poly inv(const poly &x) {
    int n = size(x);
    if (n == 1)
        return poly{x[0].inv()};
    auto c = inv(pre(x, (n + 1) / 2));
    return pre(c * (poly{2} - c * x), n);
}

poly log(const poly &x) {
    int n = size(x);
    assert(x[0].val() == 1);
    return pre(integrate(diff(x) * inv(x)), n);
}

poly exp(const poly &x) {
    assert(x[0].val() == 0);
    int n = size(x);
    if (n == 1)
        return poly{1};
    auto c = exp(pre(x, (n + 1) / 2));
    return pre(c * (poly{1} - log(pre(c, n)) + x), n);
}

pair<poly, poly> divmod(const poly &a, const poly &b) {
    assert(!b.empty());
    if (b.back().val() == 0)
        return divmod(a, shrink(b));
    if (a.empty())
        return make_pair(poly{}, poly{});
    if (a.back().val() == 0)
        return divmod(shrink(a), b);
    int n = max(0, size(a) - size(b) + 1);
    if (n == 0)
        return make_pair(poly{}, a);
    auto c = a;
    auto d = b;
    reverse(c.begin(), c.end());
    reverse(d.begin(), d.end());
    d.resize(n);
    c *= inv(d);
    c.resize(n);
    reverse(c.begin(), c.end());
    return make_pair(c, pre(a - c * b, (int)b.size() - 1));
}

poly shift(poly a, mint c) {
    const int n = a.size();
    poly e(n);
    mint tmp = 1;
    for (int i = 0; i < n; ++i) {
        e[n - 1 - i] = tmp * fact_inv(i);
        tmp *= c;
    }
    for (int i = 0; i < n; ++i) {
        a[i] *= fact(i);
    }
    a *= e;
    poly res(n);
    for (int i = 0; i < n; ++i) {
        res[i] = a[i + n - 1] * fact_inv(i);
    }
    return res;
}

mint eval(const poly &a, const mint &x) {
    int n = a.size();
    mint ans = 0;
    for (int i = n - 1; i >= 0; --i) {
        ans = ans * x + a[i];
    }
    return ans;
}

poly multipoint_evalution(const poly &a, const poly &b) {
    int n = b.size();
    vector<poly> v(n * 2);
    rep(i, 0, n) { v[i + n] = poly{-mint(b[i]), mint(1)}; }
    for (int i = n - 1; i >= 1; --i) {
        v[i] = v[i * 2] * v[i * 2 + 1];
    }
    poly ans(n);
    v[0] = a;
    rep(i, 1, n * 2) {
        v[i] = divmod(v[i / 2], v[i]).second;
        if (i >= n)
            ans[i - n] = v[i][0];
    }
    return ans;
}

mint difference_product(poly a) {
    reverse(a.begin(), a.end());
    int n = a.size();
    vector<poly> v(n * 2), tb2(n * 2);
    vector<poly> tb(n * 2);
    int sz = 1;
    while (sz < n)
        sz *= 2;
    rep(i, 0, n) { v[(i + sz - n) % n + n] = poly{-a[i], mint(1)}; }
    rep(i, 0, n) { tb[(i + sz - n) % n + n] = poly{a[i], -mint(1)}; }
    for (int i = n - 1; i >= 1; --i) {
        v[i] = v[i * 2] * v[i * 2 + 1];
    }
    for (int i = n - 1; i >= 1; --i) {
        tb[i] = tb[i * 2] * tb[i * 2 + 1];
    }
    mint ans = 1;
    tb2[1] = poly{1};
    auto dfs = [&](auto dfs, lint i) -> void {
        if (i >= n) {
            return;
        }
        tb2[i * 2] = divmod(tb2[i], v[i * 2]).second;
        dfs(dfs, i * 2);
        tb2[i * 2 + 1] = divmod(tb2[i] * tb[i * 2], v[i * 2 + 1]).second;
        dfs(dfs, i * 2 + 1);
    };
    dfs(dfs, 1);
    rep(i, 0, n) {
        // cerr<<tb2[i+n].size()<<endl;
        ans *= tb2[i + n].size() ? tb2[i + n][0] : mint(0);
    }

    return ans;
}

vector<mint> composition(vector<mint> f, vector<mint> g) {
    int n = f.size(), m = g.size();
    assert(n == m);
    vector<mint> res(n);
    int b = ceil(sqrt(n));
    vector<vector<mint>> g_pow(b + 1);
    g_pow[0] = vector<mint>{1};
    for (int i = 0; i < b; ++i) {
        g_pow[i + 1] = g_pow[i] * g;
        g_pow[i + 1].resize(n);
    }
    vector<mint> g_pow2 = vector<mint>{1};
    for (int i = 0; i < n; i += b) {
        vector<mint> tmp;
        for (int j = i; j < std::min(i + b, n); ++j) {
            tmp += g_pow[j - i] * f[j];
        }
        res += tmp * g_pow2;
        res.resize(n);
        g_pow2 *= g_pow[b];
        g_pow2.resize(n);
    }
    return res;
}
#line 6 "main.cpp"
using namespace atcoder;

mint solve(vec p) {
    lint n = p.size();
    // vector<vector<mint>>v(n,vector<mint>(n));
    // rep(i,n)rep(j,n){
    //     v[i][j]=comb(p[i]+i-1+j,j);
    // }
    // output2(v);
    mint ans = 1;
    vector<pair<poly, poly>> v(n * 2);
    rep(i, n) v[i + n] = make_pair(poly{i % 2 ? -1 : 1}, poly{p[i] + i, 1});
    for (int i = n - 1; i >= 1; --i) {
        v[i] = make_pair(v[i * 2].first * v[i * 2 + 1].second +
                             v[i * 2 + 1].first * v[i * 2].second,
                         v[i * 2].second * v[i * 2 + 1].second);
    }
    // cout<<v[1].second.size()<<endl;
    // output(v[1].first);
    poly q(n);
    rep(i, n) q[i] = -(p[i] + i);
    auto res = multipoint_evalution(v[1].first, q);
    rep(i, n) ans *= res[i];
    rep(i, 1, n) { ans /= fact(i) * fact(i); }
    // rep(i,n){
    //     rep(j,n){
    //         if(v[i][i].val()==0&&v[j][i].val()!=0){
    //             swap(v[i],v[j]);
    //             ans*=-1;
    //             break;
    //         }
    //     }
    //     if(v[i][i].val()==0){
    //         return mint(0);
    //     }
    //     rep(j,n){
    //         if(i==j)continue;
    //         mint tmp=v[j][i]/v[i][i];
    //         rep(k,n){
    //             v[j][k]-=v[i][k]*tmp;
    //         }
    //     }
    //     // output2(v);
    // }
    // rep(i,n)ans*=v[i][i];
    return ans;
}

int main(int argc, char *argv[]) {
    registerValidation(argc, argv);
    int n = inf.readInt(1, 100000, "n");
    vector<lint> p(n);
    inf.readEoln();
    rep(i, n) {
        if (i) {
            inf.readSpace();
        }
        p[i] = inf.readInt(1, 1'000'000'000, "p_i");
    }
    inf.readEoln();
    inf.readEof();
    cout << solve(p) << endl;
}