#include<bits/stdc++.h>

#define overload4(_1, _2, _3, _4, name, ...) name
#define rep1(i, n) for (ll i = 0; i < ll(n); ++i)
#define rep2(i, s, n) for (ll i = ll(s); i < ll(n); ++i)
#define rep3(i, s, n, d) for(ll i = ll(s); i < ll(n); i+=d)
#define rep(...) overload4(__VA_ARGS__,rep3,rep2,rep1)(__VA_ARGS__)
#define rrep1(i, n) for (ll i = ll(n)-1; i >= 0; i--)
#define rrep2(i, n, t) for (ll i = ll(n)-1; i >= (ll)t; i--)
#define rrep3(i, n, t, d) for (ll i = ll(n)-1; i >= (ll)t; i-=d)
#define rrep(...) overload4(__VA_ARGS__,rrep3,rrep2,rrep1)(__VA_ARGS__)
#define all(a) a.begin(),a.end()
#define rall(a) a.rbegin(),a.rend()
#define SUM(a) accumulate(all(a),0LL)
#define MIN(a) *min_element(all(a))
#define MAX(a) *max_element(all(a))
#define SORT(a) sort(all(a));
#define REV(a) reverse(all(a));
#define SZ(a) int(a.size())
#define popcount(x) __builtin_popcountll(x)
#define pf push_front
#define pb push_back
#define ef emplace_front
#define eb emplace_back
#define ppf pop_front
#define ppb pop_back
#ifdef __LOCAL
#define debug(...) { cout << #__VA_ARGS__; cout << ": "; print(__VA_ARGS__); cout << flush; }
#else
#define debug(...) void(0);
#endif
#define INT(...) int __VA_ARGS__;scan(__VA_ARGS__)
#define LL(...) ll __VA_ARGS__;scan(__VA_ARGS__)
#define STR(...) string __VA_ARGS__;scan(__VA_ARGS__)
#define CHR(...) char __VA_ARGS__;scan(__VA_ARGS__)
#define DBL(...) double __VA_ARGS__;scan(__VA_ARGS__)
#define LD(...) ld __VA_ARGS__;scan(__VA_ARGS__)
using namespace std;
using ll = long long;
using ld = long double;
using P = pair<int, int>;
using LP = pair<ll, ll>;
using vi = vector<int>;
using vvi = vector<vi>;
using vvvi = vector<vvi>;
using vl = vector<ll>;
using vvl = vector<vl>;
using vvvl = vector<vvl>;
using vd = vector<double>;
using vvd = vector<vd>;
using vs = vector<string>;
using vc = vector<char>;
using vvc = vector<vc>;
using vb = vector<bool>;
using vvb = vector<vb>;
using vp = vector<P>;
using vvp = vector<vp>;
using vlp = vector<LP>;
using vvlp = vector<vlp>;
template<class T>
using PQ = priority_queue<pair<T, int>, vector<pair<T, int>>, greater<pair<T, int>>>;

template<class S, class T>
istream &operator>>(istream &is, pair<S, T> &p) { return is >> p.first >> p.second; }

template<class S, class T>
ostream &operator<<(ostream &os, const pair<S, T> &p) { return os << '{' << p.first << ", " << p.second << '}'; }

template<class S, class T, class U>
istream &operator>>(istream &is, tuple<S, T, U> &t) { return is >> get<0>(t) >> get<1>(t) >> get<2>(t); }

template<class S, class T, class U>
ostream &operator<<(ostream &os, const tuple<S, T, U> &t) {
    return os << '{' << get<0>(t) << ", " << get<1>(t) << ", " << get<2>(t) << '}';
}

template<class T>
istream &operator>>(istream &is, vector<T> &v) {
    for (T &t: v) { is >> t; }
    return is;
}

template<class T>
ostream &operator<<(ostream &os, const vector<T> &v) {
    os << '[';
    rep(i, v.size()) os << v[i] << (i == int(v.size() - 1) ? "" : ", ");
    return os << ']';
}

template<class T>
ostream &operator<<(ostream &os, const deque<T> &v) {
    os << '[';
    rep(i, v.size()) os << v[i] << (i == int(v.size() - 1) ? "" : ", ");
    return os << ']';
}

template<class T>
ostream &operator<<(ostream &os, const set<T> &st) {
    os << '{';
    auto it = st.begin();
    while (it != st.end()) {
        os << (it == st.begin() ? "" : ", ") << *it;
        it++;
    }
    return os << '}';
}

template<class T>
ostream &operator<<(ostream &os, const multiset<T> &st) {
    os << '{';
    auto it = st.begin();
    while (it != st.end()) {
        os << (it == st.begin() ? "" : ", ") << *it;
        it++;
    }
    return os << '}';
}

template<class T, class U>
ostream &operator<<(ostream &os, const map<T, U> &mp) {
    os << '{';
    auto it = mp.begin();
    while (it != mp.end()) {
        os << (it == mp.begin() ? "" : ", ") << *it;
        it++;
    }
    return os << '}';
}

template<class T>
void vecout(const vector<T> &v, char div = '\n') {
    rep(i, v.size()) cout << v[i] << (i == int(v.size() - 1) ? '\n' : div);
}

template<class T>
bool constexpr chmin(T &a, T b) {
    if (a > b) {
        a = b;
        return true;
    }
    return false;
}

template<class T>
bool constexpr chmax(T &a, T b) {
    if (a < b) {
        a = b;
        return true;
    }
    return false;
}

void scan() {}

template<class Head, class... Tail>
void scan(Head &head, Tail &... tail) {
    cin >> head;
    scan(tail...);
}

template<class T>
void print(const T &t) { cout << t << '\n'; }

template<class Head, class... Tail>
void print(const Head &head, const Tail &... tail) {
    cout << head << ' ';
    print(tail...);
}

template<class... T>
void fin(const T &... a) {
    print(a...);
    exit(0);
}

template<class T>
vector<T> &operator+=(vector<T> &v, T x) {
    for (T &t: v) t += x;
    return v;
}

template<class T>
vector<T> &operator-=(vector<T> &v, T x) {
    for (T &t: v) t -= x;
    return v;
}

template<class T>
vector<T> &operator*=(vector<T> &v, T x) {
    for (T &t: v) t *= x;
    return v;
}

template<class T>
vector<T> &operator/=(vector<T> &v, T x) {
    for (T &t: v) t /= x;
    return v;
}

struct Init_io {
    Init_io() {
        ios::sync_with_stdio(false);
        cin.tie(nullptr);
        cout.tie(nullptr);
        cout << boolalpha << fixed << setprecision(15);
        cerr << boolalpha << fixed << setprecision(15);
    }
} init_io;

const string yes[] = {"no", "yes"};
const string Yes[] = {"No", "Yes"};
const string YES[] = {"NO", "YES"};
const int inf = 1001001001;
const ll linf = 1001001001001001001;

void rearrange(const vi &) {}

template<class T, class... Tail>
void rearrange(const vi &ord, vector<T> &head, Tail &...tail) {
    assert(ord.size() == head.size());
    vector<T> ori = head;
    rep(i, ord.size()) head[i] = ori[ord[i]];
    rearrange(ord, tail...);
}

template<class T, class... Tail>
void sort_by(vector<T> &head, Tail &... tail) {
    vi ord(head.size());
    iota(all(ord), 0);
    sort(all(ord), [&](int i, int j) { return head[i] < head[j]; });
    rearrange(ord, head, tail...);
}

bool in_rect(int i, int j, int h, int w) {
    return 0 <= i and i < h and 0 <= j and j < w;
}

template<class T, class S>
vector<T> cumsum(const vector<S> &v, bool shift_one = true) {
    int n = v.size();
    vector<T> res;
    if (shift_one) {
        res.resize(n + 1);
        rep(i, n) res[i + 1] = res[i] + v[i];
    } else {
        res.resize(n);
        if (n) {
            res[0] = v[0];
            rep(i, 1, n) res[i] = res[i - 1] + v[i];
        }
    }
    return res;
}

vvi graph(int n, int m, bool directed = false, int origin = 1) {
    vvi G(n);
    rep(_, m) {
        INT(u, v);
        u -= origin, v -= origin;
        G[u].pb(v);
        if (!directed) G[v].pb(u);
    }
    return G;
}

template<class T>
vector<vector<pair<int, T>>> weighted_graph(int n, int m, bool directed = false, int origin = 1) {
    vector<vector<pair<int, T>>> G(n);
    rep(_, m) {
        int u, v;
        T w;
        scan(u, v, w);
        u -= origin, v -= origin;
        G[u].eb(v, w);
        if (!directed) G[v].eb(u, w);
    }
    return G;
}

template<int mod>
class modint {
    ll x;
public:
    constexpr modint(ll x = 0) : x((x % mod + mod) % mod) {}

    static constexpr int get_mod() { return mod; }

    constexpr int val() const { return x; }

    constexpr modint operator-() const { return modint(-x); }

    constexpr modint &operator+=(const modint &a) {
        if ((x += a.val()) >= mod) x -= mod;
        return *this;
    }

    constexpr modint &operator++() { return *this += 1; }

    constexpr modint &operator-=(const modint &a) {
        if ((x += mod - a.val()) >= mod) x -= mod;
        return *this;
    }

    constexpr modint &operator--() { return *this -= 1; }

    constexpr modint
    &

    operator*=(const modint &a) {
        (x *= a.val()) %= mod;
        return *this;
    }

    constexpr modint

    operator+(const modint &a) const {
        modint res(*this);
        return res += a;
    }

    constexpr modint

    operator-(const modint &a) const {
        modint res(*this);
        return res -= a;
    }

    constexpr modint

    operator*(const modint &a) const {
        modint res(*this);
        return res *= a;
    }

    constexpr modint
    pow(ll
        t) const {
        modint res = 1, a(*this);
        while (t > 0) {
            if (t & 1) res *= a;
            t >>= 1;
            a *= a;
        }
        return res;
    }

    template<int m>
    friend istream &operator>>(istream &, modint<m> &);

    // for prime mod
    constexpr modint

    inv() const { return pow(mod - 2); }

    constexpr modint
    &

    operator/=(const modint &a) { return *this *= a.inv(); }

    constexpr modint operator/(const modint &a) const {
        modint res(*this);
        return res /= a;
    }

    // constraints : mod = 2 or val = 0 or val^((mod-1)/2) ≡ 1
    //               mod is prime
    // time complexity : O(log^2 p)
    // reference : https://nyaannyaan.github.io/library/modulo/mod-sqrt.hpp
    modint sqrt() const {
        if (x < 2) return x;
        assert(this->pow((mod - 1) >> 1).val() == 1);
        modint b = 1;
        while (b.pow((mod - 1) >> 1).val() == 1) b += 1;
        ll m = mod - 1, e = 0;
        while (~m & 1) m >>= 1, e += 1;
        modint X = this->pow((m - 1) >> 1);
        modint Y = (*this) * X * X;
        X *= *this;
        modint Z = b.pow(m);
        while (Y.val() != 1) {
            ll j = 0;
            modint t = Y;
            while (t.val() != 1) {
                j += 1;
                t *= t;
            }
            Z = Z.pow(1LL << (e - j - 1));
            X *= Z;
            Z *= Z;
            Y *= Z;
            e = j;
        }
        return X;
    }
};

using modint998244353 = modint<998244353>;
using modint1000000007 = modint<1000000007>;

template<int mod>
istream &operator>>(istream &is, modint<mod> &a) { return is >> a.x; }

template<int mod>
constexpr ostream &operator<<(ostream &os, const modint<mod> &a) { return os << a.val(); }

template<int mod>
constexpr bool operator==(const modint<mod> &a, const modint<mod> &b) { return a.val() == b.val(); }

template<int mod>
constexpr bool operator!=(const modint<mod> &a, const modint<mod> &b) { return a.val() != b.val(); }

template<int mod>
constexpr modint<mod> &operator++(modint<mod> &a) {
    return a += 1;
}

template<int mod>
constexpr modint<mod> &operator--(modint<mod> &a) {
    return a -= 1;
}

using mint = modint998244353;

using vm = vector<mint>;
using vvm = vector<vm>;

class NTT {
    int pr;

    constexpr ll pow_mod(ll x, ll n, int m) {
        if (m == 1) return 0;
        ll res = 1;
        ll now = x % m;
        while (n > 0) {
            if (n & 1) res = (res * now) % m;
            now = (now * now) % m;
            n >>= 1;
        }
        return res;
    }

    constexpr int primitive_root(int mod) {
        if (mod == 2) return 1;
        if (mod == 167772161) return 3;
        if (mod == 469762049) return 3;
        if (mod == 754974721) return 11;
        if (mod == 998244353) return 3;
        int divs[20] = {};
        divs[0] = 2;
        int cnt = 1;
        int x = (mod - 1) / 2;
        while (x % 2 == 0) x /= 2;
        for (int i = 3; (ll) 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(g, (mod - 1) / divs[i], mod) == 1) {
                    ok = false;
                    break;
                }
            }
            if (ok) return g;
        }
    }

public:
    NTT() { init(mint::get_mod()); }

    mint sum_e[30]; // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
    mint sum_ie[30];  // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]

    void init(int mod) {
        pr = primitive_root(mod);
        mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
        int cnt2 = __builtin_ctz(mint::get_mod() - 1);
        mint e = mint(pr).pow((mint::get_mod() - 1) >> cnt2), ie = e.inv();
        for (int i = cnt2; i >= 2; i--) {
            // e^(2^i) == 1
            es[i - 2] = e;
            ies[i - 2] = ie;
            e *= e;
            ie *= ie;
        }
        mint now = 1;
        for (int i = 0; i <= cnt2 - 2; i++) {
            sum_e[i] = es[i] * now;
            now *= ies[i];
        }
        now = 1;
        for (int i = 0; i <= cnt2 - 2; i++) {
            sum_ie[i] = ies[i] * now;
            now *= es[i];
        }
    }

    void operator()(vm &a, bool inverse = false) {
        int n = a.size();
        int h = __builtin_ctz(n);
        if (inverse) {
            rrep(ph, h + 1, 1) {
                int w = 1 << (ph - 1), p = 1 << (h - ph);
                mint now = 1;
                rep(s, w) {
                    int offset = s << (h - ph + 1);
                    rep(i, p) {
                        auto l = a[i + offset];
                        auto r = a[i + offset + p];
                        a[i + offset] = l + r;
                        a[i + offset + p] = (l - r) * now;
                    }
                    now *= sum_ie[__builtin_ctz(~(unsigned int) (s))];
                }
            }
            mint iv = mint(n).inv();
            rep(i, n) a[i] *= iv;
        } else {
            rep(ph, 1, h + 1) {
                int w = 1 << (ph - 1), p = 1 << (h - ph);
                mint now = 1;
                rep(s, w) {
                    int offset = s << (h - ph + 1);
                    rep(i, p) {
                        auto l = a[i + offset];
                        auto r = a[i + offset + p] * now;
                        a[i + offset] = l + r;
                        a[i + offset + p] = l - r;
                    }
                    now *= sum_e[__builtin_ctz(~(unsigned int) (s))];
                }
            }
        }
    }
} ntt;

class fps : public vector<mint> {
    static fps convolution(const fps &a, const fps &b) {
        if (a.empty()) return {};
        if (b.empty()) return {};
        int s = a.size() + b.size() - 1;
        if (min(a.size(), b.size()) <= 50) {
            fps res(s);
            if (a.size() >= b.size()) {
                rep(i, a.size()) rep(j, b.size()) res[i + j] += a[i] * b[j];
            } else {
                rep(j, b.size()) rep(i, a.size()) res[i + j] += a[i] * b[j];
            }
            return res;
        }
        int t = 1;
        while (t < s) t *= 2;
        fps A(t), B(t);
        rep(i, a.size()) A[i] = a[i];
        rep(i, b.size()) B[i] = b[i];
        ntt(A);
        ntt(B);
        rep(i, t) A[i] *= B[i];
        ntt(A, true);
        A.resize(s);
        return A;
    }

public:
    using vector<mint>::vector;

    mint eval(mint x) const {
        mint res = 0;
        mint now = 1;
        rep(i, this->size()) {
            res += (*this)[i] * now;
            now *= x;
        }
        return res;
    }

    fps pre(int n) const {
        return fps(this->begin(), this->begin() + min(n, (int) this->size()));
    }

    // return f'(x)
    fps differ() const {
        int n = this->size();
        fps res(n - 1);
        rep2(i, 1, n) res[i - 1] = (*this)[i] * i;
        return res;
    }

    // return ∫ f(x)dx
    fps integral() const {
        int n = this->size();
        if (n == 0) return fps();
        fps res(n + 1);
        rep(i, n) res[i + 1] = (*this)[i] / (i + 1);
        return res;
    };

    fps operator>>(int n) const {
        if ((int) this->size() <= n) return {};
        fps res(*this);
        res.erase(res.begin(), res.begin() + n);
        return res;
    }

    fps operator<<(int n) const {
        fps res(*this);
        res.insert(res.begin(), n, 0);
        return res;
    }

    fps &operator+=(const fps &a) {
        if (this->size() < a.size()) this->resize(a.size());
        rep(i, a.size()) (*this)[i] += a[i];
        return *this;
    }

    fps &operator-=(const fps &a) {
        if (this->size() < a.size()) this->resize(a.size());
        rep(i, a.size()) (*this)[i] -= a[i];
        return *this;
    }

    fps &operator*=(const fps &a) {
        return *this = fps::convolution(*this, a);
    }

    fps &operator*=(mint k) {
        rep(i, this->size()) (*this)[i] *= k;
        return *this;
    }

    fps operator+(const fps &a) const {
        fps res(*this);
        return res += a;
    }

    fps operator-(const fps &a) const {
        fps res(*this);
        return res -= a;
    }

    fps operator*(const fps &a) const {
        fps res(*this);
        return res *= a;
    }

    fps operator*(mint k) const {
        fps res(*this);
        return res *= k;
    }


//    // P /= (ax + b)
//    constexpr void divide(T a = 0, T b = 1) {
//        int n = this->size();
//        assert(n >= 2);
//        assert(a != 0 or b != 0);
//        if (b == T(0)) {
//            assert((*this)[0] == T(0));
//            T inv = T(1) / a;
//            rep(i, n - 1) (*this)[i] = (*this)[i + 1] * inv;
//            this->back() = T(0);
//        } else {
//            T inv = T(1) / b;
//            rep(i, n - 1) {
//                (*this)[i] *= inv;
//                (*this)[i + 1] -= (*this)[i] * a;
//            }
//            assert(this->back() == T(0));
//        }
//    }

    // reference of inv, log, exp, pow : https://opt-cp.com/fps-fast-algorithms/
    // time complexity : O(n log n)
    fps inv(int deg = -1) const {
        int n = this->size();
        assert(n and (*this)[0].val());
        if (deg == -1) deg = n;
        fps res(deg);
        res[0] = (*this)[0].inv();
        for (int m = 1; m < deg; m <<= 1) {
            fps f(2 * m), g(2 * m);
            rep(i, min(n, 2 * m)) f[i] = (*this)[i];
            rep(i, m) g[i] = res[i];
            ntt(f), ntt(g);
            rep(i, 2 * m) f[i] *= g[i];
            ntt(f, true);
            rep(i, m) f[i] = 0;
            ntt(f);
            rep(i, 2 * m) f[i] *= g[i];
            ntt(f, true);
            rep(i, m, min(2 * m, deg)) res[i] = -f[i];
        }
        return res;
    }

    fps &divide_inplace(const fps &a, int d = -1) {
        int n = this->size();
        if (d == -1) d = n;
        assert(d >= 0);
        *this = convolution(*this, a.inv(d));
        this->resize(d);
        return *this;
    }

    fps divide(const fps &a, int d = -1) {
        fps res(*this);
        return res.divide_inplace(a, d);
    }

    // time complexity : O(n log n)
    fps log(int deg = -1) const {
        int n = this->size();
        assert(n and (*this)[0].val() == 1);
        if (deg == -1) deg = n;
        fps res(this->differ());
        res.divide_inplace(*this, deg);
        res = res.integral();
        res.pop_back();
        return res;
    }

    // time complexity : O(n log n)
    fps exp(int deg = -1) const {
        int n = this->size();
        assert(n and (*this)[0].val() == 0);
        if (deg == -1) deg = n;
        fps g{1}, g_fft, f(*this);
        f.resize(deg);
        f[0] = 1;
        fps h_prime(this->differ());
        h_prime.pb(0);
        for (int m = 1; m < deg; m *= 2) {
            // prepare
            fps f_fft(f.begin(), f.begin() + m);
            f_fft.resize(2 * m);
            ntt(f_fft);

            // Step 2.a'
            if (m > 1) {
                fps _f(m);
                rep(i, m) _f[i] = f_fft[i] * g_fft[i];
                ntt(_f, true);
                _f.erase(_f.begin(), _f.begin() + m / 2);
                _f.resize(m), ntt(_f);
                rep(i, m) _f[i] *= g_fft[i];
                ntt(_f, true);
                _f.resize(m / 2);
                _f *= -1;
                g.insert(g.end(), _f.begin(), _f.begin() + m / 2);
            }

            // Step 2.b'--d'
            fps t(f.begin(), f.begin() + m);
            t = t.differ();
            t.pb(0);
            {
                // Step 2.b'
                fps r(h_prime.begin(), h_prime.begin() + m - 1);
                // Step 2.c'
                r.resize(m);
                ntt(r);
                rep(i, m) r[i] *= f_fft[i];
                ntt(r, true);
                // Step 2.d'
                t -= r;
                t.insert(t.begin(), t.back());
                t.pop_back();
            }

            // Step 2.e'
            t.resize(2 * m);
            ntt(t);
            g_fft = g;
            g_fft.resize(2 * m);
            ntt(g_fft);
            rep(i, 2 * m) t[i] *= g_fft[i];
            ntt(t, true);
            t.resize(m);

            // Step 2.f'
            fps v(f.begin() + m, f.begin() + min(deg, 2 * m));
            v.resize(m);
            t.insert(t.begin(), m - 1, 0);
            t.push_back(0);
            t = t.integral();
            rep(i, m) v[i] -= t[m + i];

            // Step 2.g'
            v.resize(2 * m);
            ntt(v);
            rep(i, 2 * m) v[i] *= f_fft[i];
            ntt(v, true);
            v.resize(m);

            // Step 2.h'
            rep(i, min(deg - m, m)) f[m + i] = v[i];
        }
        return f;
    }

    // time complexity : O(n log n)
    fps pow(ll k, int deg = -1) const {
        int n = this->size();
        if (deg == -1) deg = n;
        assert(k >= 0);
        if (k == 0) {
            fps res(deg);
            if (deg > 0) res[0] = 1;
            return res;
        }
        int l = 0;
        while (l < n && (*this)[l].val() == 0) ++l;
        if (l > (deg - 1) / k or l == n) return fps(deg);
        mint c = (*this)[l];
        fps res(this->begin() + l, this->end());
        res *= c.inv();
        res = res.log(deg - l * k);
        res *= k;
        res = res.exp();
        res *= c.pow(k);
        res.insert(res.begin(), l * k, 0);
        return res;
    }

    // time complexity : O(nt)  where t is the number of non-zero elements
    fps sparse_pow(ll k, int deg = -1) const {
        int n = this->size();
        if (deg == -1) deg = n;
        assert(k >= 0);
        if (deg == 0) return {};
        if (k == 0) {
            fps res(deg);
            res[0] = 1;
            return res;
        }

        int l = 0;
        while (l < n && (*this)[l].val() == 0) ++l;
        if (l > (deg - 1) / k or l == n) return fps(deg);
        deg -= l * k;

        vector<pair<int, mint>> v;
        rep(i, n) if ((*this)[i].val()) v.eb(i - l, (*this)[i]);
        fps res(deg);
        res[0] = v[0].second.pow(k);
        mint iv_v0 = v[0].second.inv();
        vm iv(deg, 1);
        rep(i, 1, deg) {
            // g = f^k
            // g'f = kgf'
            for (auto [d, coef]: v) {
                if (!d) continue;
                if (d > i) break;
                res[i] += coef * d * res[i - d];
            }
            res[i] *= k;
            for (auto [d, coef]: v) {
                if (!d) continue;
                if (d >= i) break;
                res[i] -= coef * res[i - d] * (i - d);
            }
            res[i] *= iv_v0 * iv[i];
            if (i + 1 < deg) iv[i + 1] = -iv[mint::get_mod() % (i + 1)] * (mint::get_mod() / (i + 1));
        }

        res.insert(res.begin(), l * k, 0);
        return res;
    }

    // constraints : ∃t, t^2 ≡ f_s and s is even
    //                  where s is the smallest index s.t. f_s != 0
    // time complexity : O(n log n)
    // reference : https://nyaannyaan.github.io/library/fps/fps-sqrt.hpp.html
    fps sqrt(int deg = -1) const {
        int n = this->size();
        if (deg == -1) deg = n;
        if (!n) return fps(deg);
        if ((*this)[0] == mint(0)) {
            rep(i, 1, n) {
                if ((*this)[i] != mint(0)) {
                    assert(~i & 1);
                    if (deg - i / 2 <= 0) break;
                    fps res = ((*this) >> i).sqrt(deg - i / 2);
                    res = res << (i / 2);
                    assert((int) res.size() == deg);
                    return res;
                }
            }
            return fps(deg, 0);
        }

        mint sqr = (*this)[0].sqrt();
        assert(sqr * sqr == (*this)[0]);
        fps res = {sqr};
        mint inv2 = mint(2).inv();
        for (int i = 1; i < deg; i <<= 1) {
            res = (res + this->pre(i << 1) * res.inv(i << 1)) * inv2;
        }
        return res.pre(deg);
    }

    // calc f(x + c)
    // time complexity : O(n log n)
    fps taylor_shift(mint c) const {
        int n = this->size();
        vm fact(n), ifact(n);
        fact[0] = 1;
        rep(i, 1, n) fact[i] = fact[i - 1] * i;
        ifact[n - 1] = fact[n - 1].inv();
        rrep(i, n - 1) ifact[i] = ifact[i + 1] * (i + 1);
        fps f(n), g(n);
        mint nc = 1;
        rep(i, n) {
            f[i] = (*this)[n - 1 - i] * fact[n - 1 - i];
            g[i] = nc * ifact[i];
            nc *= c;
        }
        fps h = f * g;
        fps res(n);
        rep(i, n) res[i] = ifact[i] * h[n - 1 - i];
        return res;
    }
};

class combination {
public:
    vector<mint> fact, ifact;

    combination(int n) : fact(n + 1), ifact(n + 1) {
        fact[0] = 1;
        for (int i = 1; i <= n; ++i) fact[i] = fact[i - 1] * i;
        ifact[n] = fact[n].inv();
        for (int i = n; i >= 1; --i) ifact[i - 1] = ifact[i] * i;
    }

    mint operator()(int n, int k) {
        if (k < 0 || k > n) return 0;
        return fact[n] * ifact[k] * ifact[n - k];
    }
} binom(2000000);

int main() {
    INT(n, m);
    vi b(n);
    scan(b);
    vi v(n - 1), mn_x(n - 1), mn_y(n - 1);
    rep(i, n - 1) {
        v[i] = b[i + 1] - b[i]; // type 2 - type 1 at i
        mn_x[i] = (v[i] >= 0 ? 0 : -v[i]);
        mn_y[i] = mn_x[i] + v[i];
    }
    int x = SUM(mn_x);
    int y = SUM(mn_y);
    if (x + y > m or x > b[0]) fin(0);
    int rem = b[0] - x;
    assert(b[n - 1] - y == rem);
    if (x + y + rem > m) fin(0);
    debug(x, y, rem);
    debug(mn_x);
    debug(mn_y);
    fps f(rem + 1);
    f[0] = 1;
    rep(i, n - 1) {
        fps g(rem + 1);
        rep(j, rem + 1) g[j] = binom.ifact[mn_x[i] + j] * binom.ifact[mn_y[i] + j];
        f *= g;
        f.resize(rem + 1);
    }
    mint sum;
    rep(i, rem + 1) {
        if (x + y + rem + i > m) break;
        mint ans = f[i];
        debug(ans);
        ans *= binom(m, x + y + rem + i);
        ans *= binom(x + y + rem + i, x + y + 2 * i);
        ans *= binom(x + y + 2 * i, x + i);
        ans *= binom.fact[x + i];
        ans *= binom.fact[y + i];
        sum += ans;
        debug(i, ans);
    }
    fin(sum);
}