// >>> TEMPLATES
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using ld = long double;
using i32 = int32_t;
using i64 = int64_t;
using u32 = uint32_t;
using u64 = uint64_t;
#define int ll
using pii = pair<int, int>;
#define overload3(_1, _2, _3, f, ...) f
#define v1_rep(n) for (int _n = (n); _n--; )
#define v2_rep(i, n) if (const int _n = (n); true) for (int i = 0; i < _n; i++)
#define v3_rep(i, a, b) if (const int _b = (b); true) for (int i = (a); i < _b; i++)
#define rep(...) overload3(__VA_ARGS__, v3_rep, v2_rep, v1_rep)(__VA_ARGS__)
#define v2_repR(i, n) for (int i = (int)(n)-1; i >= 0; i--)
#define v3_repR(i, a, b) if (const int _a = (a); true) for (int i = int(b)-1; i >= _a; i--)
#define repR(...) overload3(__VA_ARGS__, v3_repR, v2_repR, v1_rep)(__VA_ARGS__)
#define rep1(i, n) if (const int _rep_n = n; true) for (int i = 1; i <= _rep_n; i++)
#define rep1R(i, n) for (int i = (int)(n); i >= 1; i--)
#define loop(i, a, B) for (int i = a; i B; i++)
#define loopR(i, a, B) for (int i = a; i B; i--)
#define all(x) begin(x), end(x)
#define allR(x) rbegin(x), rend(x)
#define pb push_back
#define eb emplace_back
#define fst first
#define snd second
template <class Int> auto constexpr inf_ = numeric_limits<Int>::max()/2-1;
auto constexpr INF32 = inf_<int32_t>;
auto constexpr INF64 = inf_<int64_t>;
auto constexpr INF   = inf_<int>;
#ifdef LOCAL
#include "debug.hpp"
#define oj_local(x, y) (y)
#else
#define dump(...) ([]() { return false; }())
#define if_debug if (0)
#define oj_local(x, y) (x)
#endif

template <class T, class Comp> struct pque : priority_queue<T, vector<T>, Comp> {
    vector<T> &data() { return this->c; }
    void clear() { this->c.clear(); }
};
template <class T> using pque_max = pque<T, less<T>>;
template <class T> using pque_min = pque<T, greater<T>>;

template <class T, class S> ostream& operator<<(ostream& os, pair<T, S> const& p) {
    return os << p.first << " " << p.second;
}
template <class T, class S> istream& operator>>(istream& is, pair<T, S>& p) {
    return is >> p.first >> p.second;
}
template <class... T> ostream& operator<<(ostream& os, tuple<T...> const& t) {
    bool f = true;
    apply([&](auto&&... x) { ((os << (f ? f = false, "" : " ") << x), ...); }, t);
    return os;
}
template <class... T> istream& operator>>(istream& is, tuple<T...>& t) {
    apply([&](auto&&... x) { ((is >> x), ...); }, t);
    return is;
}
template <class T, class = typename T::iterator, enable_if_t<!is_same<T, string>::value, int> = 0>
ostream& operator<<(ostream& os, T const& a) {
    bool f = true;
    for (auto const& x : a) os << (f ? "" : " ") << x, f = false;
    return os;
}
template <class T, size_t N, enable_if_t<!is_same<T, char>::value, int> = 0>
ostream& operator<<(ostream& os, const T (&a)[N]) {
    bool f = true;
    for (auto const& x : a) os << (f ? "" : " ") << x, f = false;
    return os;
}
template <class T, class = decltype(begin(declval<T&>())), class = typename enable_if<!is_same<T, string>::value>::type>
istream& operator>>(istream& is, T &a) { for (auto& x : a) is >> x; return is; }
struct IOSetup {
    IOSetup() {
        cin.tie(nullptr);
        ios::sync_with_stdio(false);
        cout << fixed << setprecision(15);
    }
} iosetup;

template <class F> struct FixPoint : private F {
    constexpr FixPoint(F&& f) : F(forward<F>(f)) {}
    template <class... T> constexpr auto operator()(T&&... x) const {
        return F::operator()(*this, forward<T>(x)...);
    }
};
struct MakeFixPoint {
    template <class F> constexpr auto operator|(F&& f) const {
        return FixPoint<F>(forward<F>(f));
    }
};
#define def(name, ...) auto name = MakeFixPoint() | [&](auto &&name, __VA_ARGS__)

template <class F> struct FixPoint_d : private F {
    const char* const name;
    static inline int level = 0;
    constexpr FixPoint_d(F&& f, const char* name) : F(forward<F>(f)), name(name) {}
    template <class... T> constexpr auto operator()(T&&... x) const {
        if constexpr (is_same_v<void, decltype(F::operator()(*this, forward<T>(x)...))>) {
#ifdef LOCAL
            cerr << string(level, '|') << name << to_s(tuple(x...)) << '\n';
#endif
            ++level;
            F::operator()(*this, forward<T>(x)...);
            --level;
#ifdef LOCAL
            cerr << string(level, '|') << name << to_s(tuple(x...)) << " -> void" << '\n';
#endif
        } else {
#ifdef LOCAL
            cerr << string(level, '|') << name << to_s(tuple(x...)) << '\n';
#endif
            ++level;
            auto ret = F::operator()(*this, forward<T>(x)...);
            --level;
#ifdef LOCAL
            cerr << string(level, '|') << name << to_s(tuple(x...)) << " -> " << to_s(ret) << '\n';
#endif
            return ret;
        }
    }
};
struct MakeFixPoint_d {
    const char* const name;
    MakeFixPoint_d(const char* name) : name(name) {}
    template <class F> constexpr auto operator|(F&& f) const {
        return FixPoint_d<F>(forward<F>(f), name);
    }
};
#ifdef LOCAL
#define def_d(name, ...) auto name = MakeFixPoint_d(#name) | [&](auto &&name, __VA_ARGS__)
#else
#define def_d def
#endif

struct Reader {
    template <class T> T get() const{ T x; cin >> x; return x; }
    template <class T> operator T() const { return get<T>(); }
    template <class... T> void operator()(T&... args) const { ((cin >> args), ...); }
} input;

namespace impl {

template <class T, size_t d> struct mvec { using type = vector<typename mvec<T, d-1>::type>; };
template <class T> struct mvec<T, 0> { using type = T; };

template <size_t i, size_t... j>
constexpr size_t head(index_sequence<i, j...>) { return i; }

template <size_t i, size_t... j>
constexpr auto tail(index_sequence<i, j...>) { return index_sequence<j...>{}; }

template <class T, size_t... i, size_t d, class Value>
auto make_v(index_sequence<i...>, const int (&s)[d], Value const& x) {
    if constexpr (sizeof...(i) == 0) {
        return (T)x;
    } else {
        typename mvec<T, sizeof...(i)>::type ret;
        auto idx = index_sequence<i...>{};
        int n = s[head(idx)];
        if (n < 0) cerr << "[error] negative size: {" << s << "}\n", abort();
        ret.reserve(n);
        while (n-- > 0) ret.emplace_back(make_v<T>(tail(idx), s, x));
        return ret;
    }
}
} // namespace impl

template <class T, size_t d = 1> using mvec = typename impl::mvec<T, d>::type;

template <class T, size_t d, class Value> auto make_v(const int (&s)[d], Value const& x) {
    return impl::make_v<T>(make_index_sequence<d>{}, s, x);
}

template <class T, size_t d> auto make_v(const int (&s)[d]) {
    return impl::make_v<T>(make_index_sequence<d>{}, s, T{});
}

template <class T, class... Size> auto make_v(Size... s) {
    return impl::make_v<T>(make_index_sequence<sizeof...(s)>{}, (int[sizeof...(s)]){s...}, T{});
}

template <class T> void quit(T const& x) { cout << x << '\n'; exit(0); }
template <class T, class U> constexpr bool chmin(T& x, U const& y) {
    return x > y ? x = y, true : false;
}
template <class T, class U> constexpr bool chmax(T& x, U const& y) {
    return x < y ? x = y, true : false;
}
template <class It> constexpr auto sumof(It b, It e) {
    return accumulate(b, e, typename iterator_traits<It>::value_type{});
}
template <class T, class = decltype(begin(declval<T&>()))>
constexpr auto min(T const& a) { return *min_element(begin(a), end(a)); }
template <class T, class = decltype(begin(declval<T&>()))>
constexpr auto max(T const& a) { return *max_element(begin(a), end(a)); }
template <class T> constexpr T min(set<T> const& st) { assert(st.size()); return *st.begin(); }
template <class T> constexpr T max(set<T> const& st) { assert(st.size()); return *prev(st.end()); }
template <class T> constexpr T min(multiset<T> const& st) { assert(st.size()); return *st.begin(); }
template <class T> constexpr T max(multiset<T> const& st) { assert(st.size()); return *prev(st.end()); }
constexpr ll max(signed x, ll y) { return max<ll>(x, y); }
constexpr ll max(ll x, signed y) { return max<ll>(x, y); }
constexpr ll min(signed x, ll y) { return min<ll>(x, y); }
constexpr ll min(ll x, signed y) { return min<ll>(x, y); }
template <class T> int sz(T const& x) { return x.size(); }
template <class C, class T>
int lbd(C const& v, T const& x) { return lower_bound(begin(v), end(v), x) - begin(v); }
template <class C, class T>
int ubd(C const& v, T const& x) { return upper_bound(begin(v), end(v), x) - begin(v); }
constexpr ll mod(ll x, ll m) { assert(m > 0); return (x %= m) < 0 ? x+m : x; }
constexpr ll div_floor(ll x, ll y) { assert(y != 0); return x/y - ((x^y) < 0 and x%y); }
constexpr ll div_ceil(ll x, ll y) { assert(y != 0); return x/y + ((x^y) > 0 and x%y); }
constexpr int dx[] = { 1, 0, -1, 0, 1, -1, -1, 1 };
constexpr int dy[] = { 0, 1, 0, -1, 1, 1, -1, -1 };
vector<int> iota(int n) {
    vector<int> idx(n);
    iota(begin(idx), end(idx), 0);
    return idx;
}
template <class Comp> vector<int> iota(int n, Comp comp) {
    vector<int> idx(n);
    iota(begin(idx), end(idx), 0);
    stable_sort(begin(idx), end(idx), comp);
    return idx;
}
constexpr int popcnt(ll x) { return __builtin_popcountll(x); }
mt19937_64 seed_{random_device{}()};
template <class Int> Int rand(Int a, Int b) { return uniform_int_distribution<Int>(a, b)(seed_); }
i64 irand(i64 a, i64 b) { return rand<i64>(a, b); } // [a, b]
u64 urand(u64 a, u64 b) { return rand<u64>(a, b); } //
template <class It> void shuffle(It l, It r) { shuffle(l, r, seed_); }
template <class V> V &operator--(V &v) { for (auto &x : v) --x; return v; }
template <class V> V &operator++(V &v) { for (auto &x : v) ++x; return v; }
bool next_product(vector<int> &v, int m) {
    repR (i, v.size()) if (++v[i] < m) return true; else v[i] = 0;
    return false;
}
bool next_product(vector<int> &v, vector<int> const& s) {
    repR (i, v.size()) if (++v[i] < s[i]) return true; else v[i] = 0;
    return false;
}
template <class Vec> int sort_unique(Vec &v) {
    sort(begin(v), end(v));
    v.erase(unique(begin(v), end(v)), end(v));
    return v.size();
}
template <class Vec, class Comp> int sort_unique(Vec &v, Comp comp) {
    sort(begin(v), end(v), comp);
    v.erase(unique(begin(v), end(v)), end(v));
    return v.size();
}
template <class It> auto prefix_sum(It l, It r) {
    vector<typename It::value_type> s = { 0 };
    while (l != r) s.emplace_back(s.back() + *l++);
    return s;
}
template <class It> auto suffix_sum(It l, It r) {
    vector<typename It::value_type> s = { 0 };
    while (l != r) s.emplace_back(*--r + s.back());
    reverse(s.begin(), s.end());
    return s;
}
template <class T> T pop(vector<T> &a) { auto x = a.back(); a.pop_back(); return x; }
template <class T> T pop_back(vector<T> &a) { auto x = a.back(); a.pop_back(); return x; }
template <class T, class V, class C> T pop(priority_queue<T, V, C> &a) { auto x = a.top(); a.pop(); return x; }
template <class T> T pop(queue<T> &a) { auto x = a.front(); a.pop(); return x; }
template <class T> T pop_front(deque<T> &a) { auto x = a.front(); a.pop_front(); return x; }
template <class T> T pop_back(deque<T> &a) { auto x = a.back(); a.pop_back(); return x; }
template <class T> T pop_front(set<T> &a) { auto x = *a.begin(); a.erase(a.begin()); return x; }
template <class T> T pop_back(set<T> &a) { auto it = prev(a.end()); auto x = *it; a.erase(it); return x; }
template <class T> T pop_front(multiset<T> &a) { auto it = a.begin(); auto x = *it; a.erase(it); return x; }
template <class T> T pop_back(multiset<T> &a) { auto it = prev(a.end()); auto x = *it; a.erase(it); return x; }
template <class A, class B>
pair<vector<A>, vector<B>> unzip(vector<pair<A, B>> const& c) {
    vector<A> a;
    vector<B> b;
    for (auto const& [x, y] : c) {
        a.push_back(x);
        b.push_back(y);
    }
    return { a, b };
}
template <class A, class B>
pair<vector<A>, vector<B>> unzip(map<A, B> const& c) {
    vector<A> a;
    vector<B> b;
    for (auto const& [x, y] : c) {
        a.push_back(x);
        b.push_back(y);
    }
    return { a, b };
}
template <class T, class... Int> auto read_v(Int... s) {
    auto x = make_v<T>(s...);
    cin >> x;
    return x;
}

template <class T> auto read_v() {
    T x;
    cin >> x;
    return x;
}

template <class... T, class... Int>
auto read(Int... s) {
    if constexpr (sizeof...(T) >= 2) {
        return read_v<tuple<T...>>(s...);
    } else {
        return read_v<T...>(s...);
    }
}

template <class T, size_t d, class... Int>
auto read(Int... s) {
    return read_v<array<T, d>>(s...);
}
// <<<
constexpr int64_t MOD = 998244353;
// >>> modint, mod table
// >>> modint
template <uint32_t md> class modint {
    static_assert(md < (1u<<31), "");
    using M = modint;
    using i64 = int64_t;
    uint32_t x;
public:
    static constexpr uint32_t mod = md;
    constexpr modint(i64 x = 0) : x((x%=md) < 0 ? x+md : x) { }
    constexpr i64 val() const { return x; }
    constexpr explicit operator i64() const { return x; }
    constexpr bool operator==(M r) const { return x == r.x; }
    constexpr bool operator!=(M r) const { return x != r.x; }
    constexpr M operator+() const { return *this; }
    constexpr M operator-() const { return M()-*this; }
    constexpr M& operator+=(M r) { x += r.x; x = (x < md ? x : x-md); return *this; }
    constexpr M& operator-=(M r) { x += md-r.x; x = (x < md ? x : x-md); return *this; }
    constexpr M& operator*=(M r) { x = (uint64_t(x)*r.x)%md; return *this; }
    constexpr M& operator/=(M r) { return *this *= r.inv(); }
    constexpr M operator+(M r) const { return M(*this) += r; }
    constexpr M operator-(M r) const { return M(*this) -= r; }
    constexpr M operator*(M r) const { return M(*this) *= r; }
    constexpr M operator/(M r) const { return M(*this) /= r; }
    friend constexpr M operator+(i64 x, M y) { return M(x)+y; }
    friend constexpr M operator-(i64 x, M y) { return M(x)-y; }
    friend constexpr M operator*(i64 x, M y) { return M(x)*y; }
    friend constexpr M operator/(i64 x, M y) { return M(x)/y; }
    constexpr M inv() const { assert(x > 0); return pow(md-2); }
    constexpr M pow(i64 n) const {
//        assert(not (x == 0 and n == 0));
        if (n < 0) return inv().pow(-n);
        M v = *this, r = 1;
        for (; n > 0; n >>= 1, v *= v) if (n&1) r *= v;
        return r;
    }
#ifdef LOCAL
    friend string to_s(M r) { return to_s(r.val(), mod); }
#endif
    friend ostream& operator<<(ostream& os, M r) { return os << r.val(); }
    friend istream& operator>>(istream& is, M &r) { i64 x; is >> x; r = x; return is; }
};
// <<<
using mint = modint<MOD>;
mint sign(int n) { return n & 1 ? -1 : +1; }
// >>> mod table
template <uint32_t mod> struct ModTable {
    static_assert(mod > 1, "");
    vector<uint32_t> fact = { 1, 1 }, finv = { 1, 1 }, inv = { 0, 1 };
    ModTable(int n = 0) {
        calc(n);
    }
    void calc(int n) {
        const int now = fact.size();
        if (n < now) return;
        assert(n < int(1e9));
        int nxt = now;
        do nxt <<= 1; while (nxt <= n);
        fact.resize(nxt);
        finv.resize(nxt);
        inv.resize(nxt);
        for (uint32_t i = now; i < nxt; i++) {
            fact[i] = uint64_t(fact[i-1]) * i % mod;
            inv[i] = mod - uint64_t(inv[mod%i]) * (mod/i) % mod;
            finv[i] = uint64_t(finv[i-1]) * inv[i] % mod;
        }
    }
};

ModTable<MOD> mod_tab;

modint<MOD> fact(int n) {
    assert(0 <= n);
    mod_tab.calc(n);
    return mod_tab.fact[n];
}
modint<MOD> finv(int n) {
    assert(0 <= n);
    mod_tab.calc(n);
    return mod_tab.finv[n];
}
modint<MOD> inv(int n) {
    assert(0 <= n);
    mod_tab.calc(n);
    return mod_tab.inv[n];
}
modint<MOD> comb(int n, int k) {
    if (k < 0) {
        return 0;
    } else if (n >= k) {
        mod_tab.calc(n);
        return (uint64_t)mod_tab.finv[k] * mod_tab.finv[n-k] % MOD * mod_tab.fact[n];
    } else if (n < 0) {
        n = -n+k-1;
        mod_tab.calc(n);
        int ans = (uint64_t)mod_tab.finv[k] * mod_tab.finv[n-k] % MOD * mod_tab.fact[n];
        if (k & 1) ans = -ans;
        return ans;
    } else {
        return 0;
    }
}
// <<<
// <<<
// >>> NTT
template <class ModInt, int64_t g>
struct NTT {
    using modint = ModInt;
    static constexpr int64_t mod = ModInt::mod, gen = g, max_lg = __builtin_ctzll(mod-1);
    // mod:prime, g:primitive root
    static_assert(mod > 0 && g > 0 && max_lg > 0, "");

    using arr_t = array<ModInt, max_lg+1>;
    static inline const arr_t ws = []() {
        arr_t ret;
        for (int i = 0; i <= max_lg; i++) {
            ret[i] = -ModInt(g).pow((mod-1)>>(i+2));
        }
        return ret;
    } ();
    static inline const arr_t iws = []() {
        arr_t ret;
        for (int i = 0; i <= max_lg; i++) {
            ret[i] = -ModInt(1)/ModInt(g).pow((mod-1)>>(i+2));
        }
        return ret;
    } ();

    static void ntt(ModInt a[], int lg) {
        for (int b = lg-1; b >= 0; b--) {
            ModInt w = 1;
            for (int i = 0, k = 0; i < (1<<lg); i += 1<<(b+1)) {
                for (int j = i; j < (i|(1<<b)); j++) {
                    const int k = j|(1<<b);
                    const auto x = a[j], y = a[k];
                    a[j] = x + y*w;
                    a[k] = x - y*w;
                }
                w *= ws[__builtin_ctz(++k)];
            }
        }
//        bit_reverse(a, 1<<lg);
    }
    static void intt(ModInt a[], int lg) {
//        bit_reverse(a, 1<<lg);
        for (int b = 0; b < lg; b++) {
            ModInt w = 1;
            for (int i = 0, k = 0; i < (1<<lg); i += 1<<(b+1)) {
                for (int j = i; j < (i|(1<<b)); j++) {
                    const int k = j|(1<<b);
                    const auto x = a[j], y = a[k];
                    a[j] = x + y;
                    a[k] = w*(x - y);
                }
                w *= iws[__builtin_ctz(++k)];
            }
        }
    }
    template <class T>
    static vector<ModInt> conv(vector<T> const& a, vector<T> const& b) {
        if (a.empty() || b.empty()) return {};
        const int s = a.size() + b.size() - 1, lg = __lg(2*s-1);
        assert(lg <= max_lg);

        vector<ModInt> aa(1<<lg);
        rep (i, a.size()) aa[i] = (int64_t)a[i];
        ntt(aa.data(), lg);

        vector<ModInt> bb(1<<lg);
        rep (i, b.size()) bb[i] = (int64_t)b[i];
        ntt(bb.data(), lg);

        const auto x = ModInt(1)/ModInt(1<<lg);
        rep (i, 1<<lg) aa[i] *= bb[i]*x;
        intt(aa.data(), lg); aa.resize(s);
        return aa;
    }
    template <class T>
    static vector<ModInt> conv(vector<T> const& a) {
        if (a.empty()) return {};
        const int s = a.size()*2 - 1, lg = __lg(2*s-1);
        assert(lg <= max_lg);

        vector<ModInt> aa(1<<lg);
        rep (i, a.size()) aa[i] = (int64_t)a[i];
        ntt(aa.data(), lg);

        const auto x = ModInt(1)/ModInt(1<<lg);
        rep (i, 1<<lg) aa[i] *= aa[i]*x;
        intt(aa.data(), lg); aa.resize(s);
        return aa;
    }
};
// <<<
using ntt = NTT<mint, 3>;
// >>> FPS
template <class NTT>
struct FormalPowerSeries : NTT, vector<typename NTT::modint> {
    using mint = typename NTT::modint;
    using NTT::conv;
    using vector<mint>::vector; // inherit constructors
    using FPS = FormalPowerSeries;
    FormalPowerSeries() : vector<mint>() {}
    FormalPowerSeries(vector<mint> const& v) : vector<mint>(v) {}
    FormalPowerSeries(mint const& x) : vector<mint>({x}) {}
    void shrink() { while (this->size() and this->back() == 0) this->pop_back(); }
    mint get(int i) const {
        assert(i >= 0);
        if (i < (int)this->size()) return (*this)[i];
        else return 0;
    }
    mint &set(int i, mint x) {
        assert(i >= 0);
        if (i >= (int)this->size()) this->resize(i+1);
        return (*this)[i] = x;
    }
    bool operator==(FPS const& r) const {
        const int n = min(this->size(), r.size());
        rep (i, n) {
            if ((*this)[i] != r[i]) return false;
        }
        for (int i = n; i < (int)this->size(); ++i) {
            if ((*this)[i] != mint(0)) return false;
        }
        for (int i = n; i < (int)r.size(); ++i) {
            if (r[i] != mint(0)) return false;
        }
        return true;
    }
    bool operator!=(FPS const& r) const { return !((*this) == r); }
    FPS operator+(FPS const& r) const { return FPS(*this) += r; }
    FPS operator-(FPS const& r) const { return FPS(*this) -= r; }
    FPS& operator+=(FPS const& r) {
        if (r.size() > this->size()) this->resize(r.size());
        rep (i, r.size()) (*this)[i] += r[i];
        return *this;
    }
    FPS& operator-=(FPS const& r) {
        if (r.size() > this->size()) this->resize(r.size());
        rep (i, r.size()) (*this)[i] -= r[i];
        return *this;
    }
    FPS operator*(FPS const& r) const {
        if (this->empty() || r.empty()) return {};
        return conv(*this, r);
    }
    FPS& operator*=(FPS const& r) { return *this = *this * r; }
    friend FPS operator+(mint const& x, FPS const& f) { return FPS{x}+f; }
    friend FPS operator-(mint const& x, FPS const& f) { return FPS{x}-f; }
    friend FPS operator*(mint const& x, FPS const& f) { return FPS{x}*f; }
    friend FPS operator+(FPS const& f, mint const& x) { return f+FPS{x}; }
    friend FPS operator-(FPS const& f, mint const& x) { return f-FPS{x}; }
    friend FPS operator*(FPS const& f, mint const& x) { return f*FPS{x}; }
    friend FPS operator+(FPS const& f) { return f; }
    friend FPS operator-(FPS const& f) { return FPS{}-f; }
    FPS take(int sz) const {
        FPS ret(this->begin(), this->begin() + min<int>(this->size(), sz));
        ret.resize(sz);
        return ret;
    }
    FPS inv(int sz = -1) const {
        assert(this->size()); assert((*this)[0] != mint(0));
        if (sz < 0) sz = this->size();
        FPS ret = { mint(1)/(*this)[0] };
        for (int i = 1; i < sz; i <<= 1) {
            ret = ret + ret - ret*ret*take(i<<1);
            ret.resize(i<<1);
        }
        ret.resize(sz);
        return ret;
    }
    FPS diff() const {
        FPS ret(max<int>(0, this->size()-1));
        rep (i, ret.size()) ret[i] = (*this)[i+1]*mint(i+1);
        return ret;
    }
    FPS integral() const {
        FPS ret(this->size()+1);
        ret[0] = 0;
        rep (i, this->size()) ret[i+1] = (*this)[i]/mint(i+1);
        return ret;
    }
    FPS log(int sz = -1) const {
        assert(this->size()); assert((*this)[0] == mint(1));
        if (sz < 0) sz = this->size();
        return (diff()*inv(sz)).take(sz-1).integral();
    }
    // FPS log(int sz = -1) const {
    //     assert(this->size()); assert((*this)[0] == mint(1));
    //     if (sz < 0) sz = this->size();
    //     auto ret = diff()*inv(sz);
    //     ret.resize(sz);
    //     for (int i = sz-1; i > 0; --i) ret[i] = ret[i-1]/mint(i);
    //     ret[0] = 0;
    //     return ret;
    // }
    FPS exp(int sz = -1) const {
        FPS ret = {mint(1)};
        if (this->empty()) return ret;
        assert((*this)[0] == mint(0));
        if (sz < 0) sz = this->size();
        for (int i = 1; i < sz; i <<= 1) {
            ret *= take(i<<1) + mint(1) - ret.log(i<<1);
            ret.resize(i<<1);
        }
        ret.resize(sz);
        return ret;
    }
    FPS pow(int64_t k, int sz = -1) const {
        if (sz < 0) sz = this->size();
        int deg = 0;
        while (deg < sz && (*this).get(deg) == mint(0)) ++deg;
        assert(k >= 0 || deg == 0);

        auto c = mint(1)/(*this).get(deg);
        FPS ret(sz-deg);
        rep (i, sz-deg) ret[i] = (*this).get(deg+i)*c;
        ret = (ret.log()*k).exp() * (*this).get(deg).pow(k);

        ret.resize(sz);
        for (int i = sz-1; i >= 0; --i) {
            int j = i-deg*k;
            ret[i] = (j >= 0 ? ret[j] : mint(0));
        }
        return ret;
    }
    mint eval(mint x) const {
        mint p = 1, ret = 0;
        rep (i, this->size()) {
            ret += (*this)[i] * p;
            p *= x;
        }
        return ret;
    }
};
// <<<
using FPS = FormalPowerSeries<ntt>;

int32_t main() {
    int n = input, m = input;

    FPS dp = {1};
    rep (n) {
        int a = input, b = input;
        --a;

        FPS f(m + 1);
        mint B = finv(m), A = finv(m);
        rep (j, m) B *= m + b - j, A *= m + a - j;
        rep (k, f.size()) {
            if (k > 0) {
                mint c = mint(m + k).inv();
                B *= (b - k + 1) * c;
                A *= (a - k + 1) * c;
            }
            f[k] = (B - A) * finv(k);
        }
        dump(f);

        dp *= f;
        dp.resize(m + 1);
    }
    cout << dp[m] * fact(m) << '\n';
}