#include <bits/stdc++.h>
using namespace std;
#define rep(i, n) for (int i = 0; i < (n); i++)
#define per(i, n) for (int i = (n)-1; i >= 0; i--)
#define rep2(i, l, r) for (int i = (l); i < (r); i++)
#define per2(i, l, r) for (int i = (r)-1; i >= (l); i--)
#define each(e, v) for (auto &e : v)
#define MM << " " <<
#define pb push_back
#define eb emplace_back
#define all(x) begin(x), end(x)
#define rall(x) rbegin(x), rend(x)
#define sz(x) (int)x.size()
using ll = long long;
using pii = pair<int, int>;
using pil = pair<int, ll>;
using pli = pair<ll, int>;
using pll = pair<ll, ll>;

template <typename T>
using minheap = priority_queue<T, vector<T>, greater<T>>;

template <typename T>
using maxheap = priority_queue<T>;

template <typename T>
bool chmax(T &x, const T &y) {
    return (x < y) ? (x = y, true) : false;
}

template <typename T>
bool chmin(T &x, const T &y) {
    return (x > y) ? (x = y, true) : false;
}

template <typename T>
int flg(T x, int i) {
    return (x >> i) & 1;
}

template <typename T>
void print(const vector<T> &v, T x = 0) {
    int n = v.size();
    for (int i = 0; i < n; i++) cout << v[i] + x << (i == n - 1 ? '\n' : ' ');
    if (v.empty()) cout << '\n';
}

template <typename T>
void printn(const vector<T> &v, T x = 0) {
    int n = v.size();
    for (int i = 0; i < n; i++) cout << v[i] + x << '\n';
}

template <typename T>
int lb(const vector<T> &v, T x) {
    return lower_bound(begin(v), end(v), x) - begin(v);
}

template <typename T>
int ub(const vector<T> &v, T x) {
    return upper_bound(begin(v), end(v), x) - begin(v);
}

template <typename T>
void rearrange(vector<T> &v) {
    sort(begin(v), end(v));
    v.erase(unique(begin(v), end(v)), end(v));
}

template <typename T>
vector<int> id_sort(const vector<T> &v, bool greater = false) {
    int n = v.size();
    vector<int> ret(n);
    iota(begin(ret), end(ret), 0);
    sort(begin(ret), end(ret), [&](int i, int j) { return greater ? v[i] > v[j] : v[i] < v[j]; });
    return ret;
}

template <typename S, typename T>
pair<S, T> operator+(const pair<S, T> &p, const pair<S, T> &q) {
    return make_pair(p.first + q.first, p.second + q.second);
}

template <typename S, typename T>
pair<S, T> operator-(const pair<S, T> &p, const pair<S, T> &q) {
    return make_pair(p.first - q.first, p.second - q.second);
}

template <typename S, typename T>
istream &operator>>(istream &is, pair<S, T> &p) {
    S a;
    T b;
    is >> a >> b;
    p = make_pair(a, b);
    return is;
}

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

struct io_setup {
    io_setup() {
        ios_base::sync_with_stdio(false);
        cin.tie(NULL);
        cout << fixed << setprecision(15);
    }
} io_setup;

const int inf = (1 << 30) - 1;
const ll INF = (1LL << 60) - 1;
// const int MOD = 1000000007;
const int MOD = 998244353;

template <int mod>
struct Mod_Int {
    int x;

    Mod_Int() : x(0) {}

    Mod_Int(long long y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}

    static int get_mod() { return mod; }

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

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

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

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

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

    Mod_Int operator++(int) {
        Mod_Int tmp = *this;
        ++*this;
        return tmp;
    }

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

    Mod_Int operator--(int) {
        Mod_Int tmp = *this;
        --*this;
        return tmp;
    }

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

    Mod_Int operator+(const Mod_Int &p) const { return Mod_Int(*this) += p; }

    Mod_Int operator-(const Mod_Int &p) const { return Mod_Int(*this) -= p; }

    Mod_Int operator*(const Mod_Int &p) const { return Mod_Int(*this) *= p; }

    Mod_Int operator/(const Mod_Int &p) const { return Mod_Int(*this) /= p; }

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

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

    Mod_Int inverse() const {
        assert(*this != Mod_Int(0));
        return pow(mod - 2);
    }

    Mod_Int pow(long long k) const {
        Mod_Int now = *this, ret = 1;
        for (; k > 0; k >>= 1, now *= now) {
            if (k & 1) ret *= now;
        }
        return ret;
    }

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

    friend istream &operator>>(istream &is, Mod_Int &p) {
        long long a;
        is >> a;
        p = Mod_Int<mod>(a);
        return is;
    }
};

using mint = Mod_Int<MOD>;

template <typename T>
struct Number_Theoretic_Transform {
    static int max_base;
    static T root;
    static vector<T> r, ir;

    Number_Theoretic_Transform() {}

    static void init() {
        if (!r.empty()) return;
        int mod = T::get_mod();
        int tmp = mod - 1;
        root = 2;
        while (root.pow(tmp >> 1) == 1) root++;
        max_base = 0;
        while (tmp % 2 == 0) tmp >>= 1, max_base++;
        r.resize(max_base), ir.resize(max_base);
        for (int i = 0; i < max_base; i++) {
            r[i] = -root.pow((mod - 1) >> (i + 2)); // r[i]  := 1 の 2^(i+2) 乗根
            ir[i] = r[i].inverse();                 // ir[i] := 1/r[i]
        }
    }

    static void ntt(vector<T> &a) {
        init();
        int n = a.size();
        assert((n & (n - 1)) == 0);
        assert(n <= (1 << max_base));
        for (int k = n; k >>= 1;) {
            T w = 1;
            for (int s = 0, t = 0; s < n; s += 2 * k) {
                for (int i = s, j = s + k; i < s + k; i++, j++) {
                    T x = a[i], y = w * a[j];
                    a[i] = x + y, a[j] = x - y;
                }
                w *= r[__builtin_ctz(++t)];
            }
        }
    }

    static void intt(vector<T> &a) {
        init();
        int n = a.size();
        assert((n & (n - 1)) == 0);
        assert(n <= (1 << max_base));
        for (int k = 1; k < n; k <<= 1) {
            T w = 1;
            for (int s = 0, t = 0; s < n; s += 2 * k) {
                for (int i = s, j = s + k; i < s + k; i++, j++) {
                    T x = a[i], y = a[j];
                    a[i] = x + y, a[j] = w * (x - y);
                }
                w *= ir[__builtin_ctz(++t)];
            }
        }
        T inv = T(n).inverse();
        for (auto &e : a) e *= inv;
    }

    static vector<T> convolve(vector<T> a, vector<T> b) {
        if (a.empty() || b.empty()) return {};
        int k = (int)a.size() + (int)b.size() - 1, n = 1;
        while (n < k) n <<= 1;
        a.resize(n), b.resize(n);
        ntt(a), ntt(b);
        for (int i = 0; i < n; i++) a[i] *= b[i];
        intt(a), a.resize(k);
        return a;
    }
};

template <typename T>
int Number_Theoretic_Transform<T>::max_base = 0;

template <typename T>
T Number_Theoretic_Transform<T>::root = T();

template <typename T>
vector<T> Number_Theoretic_Transform<T>::r = vector<T>();

template <typename T>
vector<T> Number_Theoretic_Transform<T>::ir = vector<T>();

using NTT = Number_Theoretic_Transform<mint>;

template <typename T>
struct Combination {
    static vector<T> _fac, _ifac;

    Combination() {}

    static void init(int n) {
        _fac.resize(n + 1), _ifac.resize(n + 1);
        _fac[0] = 1;
        for (int i = 1; i <= n; i++) _fac[i] = _fac[i - 1] * i;
        _ifac[n] = _fac[n].inverse();
        for (int i = n; i >= 1; i--) _ifac[i - 1] = _ifac[i] * i;
    }

    static T fac(int k) { return _fac[k]; }

    static T ifac(int k) { return _ifac[k]; }

    static T inv(int k) { return fac(k - 1) * ifac(k); }

    static T P(int n, int k) {
        if (k < 0 || n < k) return 0;
        return fac(n) * ifac(n - k);
    }

    static T C(int n, int k) {
        if (k < 0 || n < k) return 0;
        return fac(n) * ifac(n - k) * ifac(k);
    }

    // k 個の区別できない玉を n 個の区別できる箱に入れる場合の数
    static T H(int n, int k) {
        if (n < 0 || k < 0) return 0;
        return k == 0 ? 1 : C(n + k - 1, k);
    }

    // n 個の区別できる玉を、k 個の区別しない箱に、各箱に 1 個以上玉が入るように入れる場合の数
    static T second_stirling_number(int n, int k) {
        T ret = 0;
        for (int i = 0; i <= k; i++) {
            T tmp = C(k, i) * T(i).pow(n);
            ret += ((k - i) & 1) ? -tmp : tmp;
        }
        return ret * ifac(k);
    }

    // n 個の区別できる玉を、k 個の区別しない箱に入れる場合の数
    static T bell_number(int n, int k) {
        if (n == 0) return 1;
        k = min(k, n);
        vector<T> pref(k + 1);
        pref[0] = 1;
        for (int i = 1; i <= k; i++) {
            if (i & 1) {
                pref[i] = pref[i - 1] - ifac(i);
            } else {
                pref[i] = pref[i - 1] + ifac(i);
            }
        }
        T ret = 0;
        for (int i = 1; i <= k; i++) ret += T(i).pow(n) * ifac(i) * pref[k - i];
        return ret;
    }
};

template <typename T>
vector<T> Combination<T>::_fac = vector<T>();

template <typename T>
vector<T> Combination<T>::_ifac = vector<T>();

using comb = Combination<mint>;

template <typename T>
vector<T> divisors(const T &n) {
    vector<T> ret;
    for (T i = 1; i * i <= n; i++) {
        if (n % i == 0) {
            ret.push_back(i);
            if (i * i != n) ret.push_back(n / i);
        }
    }
    sort(begin(ret), end(ret));
    return ret;
}

template <typename T>
vector<pair<T, int>> prime_factor(T n) {
    vector<pair<T, int>> ret;
    for (T i = 2; i * i <= n; i++) {
        int cnt = 0;
        while (n % i == 0) cnt++, n /= i;
        if (cnt > 0) ret.emplace_back(i, cnt);
    }
    if (n > 1) ret.emplace_back(n, 1);
    return ret;
}

template <typename T>
bool is_prime(const T &n) {
    if (n == 1) return false;
    for (T i = 2; i * i <= n; i++) {
        if (n % i == 0) return false;
    }
    return true;
}

// 1,2,...,n のうち k と互いに素である自然数の個数
template <typename T>
T coprime(T n, T k) {
    vector<pair<T, int>> ps = prime_factor(k);
    int m = ps.size();
    T ret = 0;
    for (int i = 0; i < (1 << m); i++) {
        T prd = 1;
        for (int j = 0; j < m; j++) {
            if ((i >> j) & 1) prd *= ps[j].first;
        }
        ret += (__builtin_parity(i) ? -1 : 1) * (n / prd);
    }
    return ret;
}

vector<bool> Eratosthenes(const int &n) {
    vector<bool> ret(n + 1, true);
    if (n >= 0) ret[0] = false;
    if (n >= 1) ret[1] = false;
    for (int i = 2; i * i <= n; i++) {
        if (!ret[i]) continue;
        for (int j = i + i; j <= n; j += i) ret[j] = false;
    }
    return ret;
}

vector<int> Eratosthenes2(const int &n) {
    vector<int> ret(n + 1);
    iota(begin(ret), end(ret), 0);
    if (n >= 0) ret[0] = -1;
    if (n >= 1) ret[1] = -1;
    for (int i = 2; i * i <= n; i++) {
        if (ret[i] < i) continue;
        for (int j = i + i; j <= n; j += i) ret[j] = min(ret[j], i);
    }
    return ret;
}

template <typename Monoid>
struct Segment_Tree {
    using F = function<Monoid(Monoid, Monoid)>;
    int n;
    vector<Monoid> seg;
    const F f;
    const Monoid e1;

    // f(f(a,b),c) = f(a,f(b,c)), f(e1,a) = f(a,e1) = a

    Segment_Tree(const vector<Monoid> &v, const F &f, const Monoid &e1) : f(f), e1(e1) {
        int m = v.size();
        n = 1;
        while (n < m) n <<= 1;
        seg.assign(2 * n, e1);
        copy(begin(v), end(v), seg.begin() + n);
        for (int i = n - 1; i > 0; i--) seg[i] = f(seg[2 * i], seg[2 * i + 1]);
    }

    Segment_Tree(int m, const Monoid &x, const F &f, const Monoid &e1) : Segment_Tree(vector<Monoid>(m, x), f, e1) {}

    void change(int i, const Monoid &x, bool update = true) {
        if (update) {
            seg[i + n] = x;
        } else {
            seg[i + n] = f(seg[i + n], x);
        }
        i += n;
        while (i >>= 1) seg[i] = f(seg[2 * i], seg[2 * i + 1]);
    }

    Monoid query(int l, int r) const {
        l = max(l, 0), r = min(r, n);
        Monoid L = e1, R = e1;
        l += n, r += n;
        while (l < r) {
            if (l & 1) L = f(L, seg[l++]);
            if (r & 1) R = f(seg[--r], R);
            l >>= 1, r >>= 1;
        }
        return f(L, R);
    }

    Monoid operator[](int i) const { return seg[n + i]; }

    template <typename C>
    int find_subtree(int i, const C &check, const Monoid &x, Monoid &M, int type) const {
        while (i < n) {
            Monoid nxt = type ? f(seg[2 * i + type], M) : f(M, seg[2 * i + type]);
            if (check(nxt, x)) {
                i = 2 * i + type;
            } else {
                M = nxt;
                i = 2 * i + (type ^ 1);
            }
        }
        return i - n;
    }

    // check((区間 [l,r] での演算結果), x) を満たす最小の r (存在しなければ n 以上の値)
    template <typename C>
    int find_first(int l, const C &check, const Monoid &x) const {
        Monoid L = e1;
        int a = l + n, b = n + n;
        while (a < b) {
            if (a & 1) {
                Monoid nxt = f(L, seg[a]);
                if (check(nxt, x)) return find_subtree(a, check, x, L, 0);
                L = nxt, a++;
            }
            a >>= 1, b >>= 1;
        }
        return n;
    }

    // check((区間 [l,r) での演算結果), x) を満たす最大の l (存在しなければ -1)
    template <typename C>
    int find_last(int r, const C &check, const Monoid &x) const {
        Monoid R = e1;
        int a = n, b = r + n;
        while (a < b) {
            if ((b & 1) || a == 1) {
                Monoid nxt = f(seg[--b], R);
                if (check(nxt, x)) return find_subtree(b, check, x, R, 1);
                R = nxt;
            }
            a >>= 1, b >>= 1;
        }
        return -1;
    }
};

struct Union_Find_Tree {
    vector<int> data;
    const int n;
    int cnt;

    Union_Find_Tree(int n) : data(n, -1), n(n), cnt(n) {}

    int root(int x) {
        if (data[x] < 0) return x;
        return data[x] = root(data[x]);
    }

    int operator[](int i) { return root(i); }

    bool unite(int x, int y) {
        x = root(x), y = root(y);
        if (x == y) return false;
        if (data[x] > data[y]) swap(x, y);
        data[x] += data[y], data[y] = x;
        cnt--;
        return true;
    }

    int size(int x) { return -data[root(x)]; }

    int count() { return cnt; };

    bool same(int x, int y) { return root(x) == root(y); }

    void clear() {
        cnt = n;
        fill(begin(data), end(data), -1);
    }
};

template <typename T>
struct Sparse_Table {
    using F = function<T(T, T)>;
    const int n;
    int height;
    vector<vector<T>> st; // st[i][j] := 区間 [j,j+2^i) での演算の結果
    vector<int> lookup;
    const F f;
    const T e;

    // f(f(a,b),c) = f(a,f(b,c)), f(e,a) = f(a,e) = a, f(a,a) = a
    // 例えば min や gcd はこれらを満たすが、+ や * は満たさない

    Sparse_Table(const vector<T> &table, const F &f, const T &e) : n((int)table.size()), f(f), e(e) {
        height = 0;
        while (n >> height) height++;
        st.assign(height, vector<T>(n));
        for (int i = 0; i < n; i++) st[0][i] = table[i];
        for (int j = 0; j < height - 1; j++) {
            for (int i = 0; i < n; i++) {
                if (i + (1 << j) < n) {
                    st[j + 1][i] = f(st[j][i], st[j][i + (1 << j)]);
                } else {
                    st[j + 1][i] = st[j][i];
                }
            }
        }
        lookup.assign(n + 1, -1);
        for (int i = 1; i <= n; i++) lookup[i] = lookup[i / 2] + 1;
    }

    T query(int l, int r) const {
        if (l >= r) return e;
        int k = lookup[r - l];
        return f(st[k][l], st[k][r - (1 << k)]);
    }

    T operator[](int i) const { return st[0][i]; }
};

template <bool directed = false>
struct Low_Link {
    struct edge {
        int to, id;
        edge(int to, int id) : to(to), id(id) {}
    };

    vector<vector<edge>> es;
    vector<int> ord, low;
    vector<bool> used;
    vector<int> articulation, bridge;
    const int n;
    int m;

    Low_Link(int n) : es(n), ord(n), low(n), used(n), n(n), m(0) {}

    void add_edge(int from, int to) {
        es[from].emplace_back(to, m);
        if (!directed) es[to].emplace_back(from, m);
        m++;
    }

    int _dfs(int now, int pre, int k) {
        used[now] = true;
        ord[now] = low[now] = k++;
        bool is_articulation = false;
        int cnt = 0;
        for (auto &e : es[now]) {
            if (e.id == pre) continue;
            if (!used[e.to]) {
                cnt++;
                k = _dfs(e.to, e.id, k);
                low[now] = min(low[now], low[e.to]);
                if (pre != -1 && low[e.to] >= ord[now]) is_articulation = true;
                if (ord[now] < low[e.to]) bridge.push_back(e.id);
            } else {
                low[now] = min(low[now], ord[e.to]);
            }
        }
        if (pre == -1 && cnt >= 2) is_articulation = true;
        if (is_articulation) articulation.push_back(now);
        return k;
    }

    void build() {
        fill(begin(used), end(used), false);
        int k = 0;
        for (int i = 0; i < n; i++) {
            if (!used[i]) k = _dfs(i, -1, k);
        }
    }
};

template <bool directed = false>
struct Biconnected_Components : Low_Link<directed> {
    using L = Low_Link<directed>;
    vector<int> comp;
    vector<bool> used;
    const int n;

    Biconnected_Components(int n) : L(n), used(n), n(n) {}

    int _dfs(int now, int pre, int top, int k) {
        used[now] = true;
        for (auto &e : this->es[now]) {
            if (comp[e.id] != -1) continue;
            if (this->ord[e.to] < this->ord[now]) {
                comp[e.id] = top;
            } else if (this->low[e.to] >= this->ord[now]) {
                comp[e.id] = k;
                k = _dfs(e.to, now, k, k + 1);
            } else {
                comp[e.id] = top;
                k = _dfs(e.to, now, top, k);
            }
        }
        return k;
    }

    int decompose() {
        this->build();
        comp.assign(this->m, -1);
        fill(begin(used), end(used), false);
        int k = 0;
        for (int i = 0; i < n; i++) {
            if (!used[i]) k = _dfs(i, -1, -1, k);
        }
        return k;
    }
};

template <typename T>
void fast_zeta_transform(vector<T> &a, bool upper) {
    int n = a.size();
    assert((n & (n - 1)) == 0);
    for (int i = 1; i < n; i <<= 1) {
        for (int j = 0; j < n; j++) {
            if (!(j & i)) {
                if (upper) {
                    a[j] += a[j | i];
                } else {
                    a[j | i] += a[j];
                }
            }
        }
    }
}

template <typename T>
void fast_mobius_transform(vector<T> &a, bool upper) {
    int n = a.size();
    assert((n & (n - 1)) == 0);
    for (int i = 1; i < n; i <<= 1) {
        for (int j = 0; j < n; j++) {
            if (!(j & i)) {
                if (upper) {
                    a[j] -= a[j | i];
                } else {
                    a[j | i] -= a[j];
                }
            }
        }
    }
}

template <typename T>
void fast_hadamard_transform(vector<T> &a, bool inverse = false) {
    int n = a.size();
    assert((n & (n - 1)) == 0);
    for (int i = 1; i < n; i <<= 1) {
        for (int j = 0; j < n; j++) {
            if (!(j & i)) {
                T x = a[j], y = a[j | i];
                a[j] = x + y, a[j | i] = x - y;
            }
        }
    }
    if (inverse) {
        T inv = T(1) / T(n);
        for (auto &e : a) e *= inv;
    }
}

template <typename T>
vector<T> bitwise_and_convolve(vector<T> a, vector<T> b) {
    int n = a.size();
    assert(b.size() == n && (n & (n - 1)) == 0);
    fast_zeta_transform(a, true), fast_zeta_transform(b, true);
    for (int i = 0; i < n; i++) a[i] *= b[i];
    fast_mobius_transform(a, true);
    return a;
}

template <typename T>
vector<T> bitwise_or_convolve(vector<T> a, vector<T> b) {
    int n = a.size();
    assert(b.size() == n && (n & (n - 1)) == 0);
    fast_zeta_transform(a, false), fast_zeta_transform(b, false);
    for (int i = 0; i < n; i++) a[i] *= b[i];
    fast_mobius_transform(a, false);
    return a;
}

template <typename T>
vector<T> bitwise_xor_convolve(vector<T> a, vector<T> b) {
    int n = a.size();
    assert(b.size() == n && (n & (n - 1)) == 0);
    fast_hadamard_transform(a), fast_hadamard_transform(b);
    for (int i = 0; i < n; i++) a[i] *= b[i];
    fast_hadamard_transform(a, true);
    return a;
}

template <typename T>
vector<T> subset_convolve(const vector<T> &a, const vector<T> &b) {
    int n = a.size();
    assert(b.size() == n && (n & (n - 1)) == 0);
    int k = __builtin_ctz(n);
    vector<vector<T>> A(k + 1, vector<T>(n, 0)), B(k + 1, vector<T>(n, 0)), C(k + 1, vector<T>(n, 0));
    for (int i = 0; i < n; i++) {
        int t = __builtin_popcount(i);
        A[t][i] = a[i], B[t][i] = b[i];
    }
    for (int i = 0; i <= k; i++) fast_zeta_transform(A[i], false), fast_zeta_transform(B[i], false);
    for (int i = 0; i <= k; i++) {
        for (int j = 0; j <= k - i; j++) {
            for (int l = 0; l < n; l++) C[i + j][l] += A[i][l] * B[j][l];
        }
    }
    for (int i = 0; i <= k; i++) fast_mobius_transform(C[i], false);
    vector<T> c(n);
    for (int i = 0; i < n; i++) c[i] = C[__builtin_popcount(i)][i];
    return c;
}

template <typename T>
void divisors_zeta_transform(vector<T> &a, bool upper) {
    int n = a.size();
    vector<bool> is_prime(n, true);
    if (!upper) {
        for (int i = 1; i < n; i++) a[0] += a[i];
    }
    for (int i = 2; i < n; i++) {
        if (!is_prime[i]) continue;
        if (upper) {
            for (int j = (n - 1) / i; j > 0; j--) {
                is_prime[j * i] = false;
                a[j] += a[j * i];
            }
        } else {
            for (int j = 1; j * i < n; j++) {
                is_prime[j * i] = false;
                a[j * i] += a[j];
            }
        }
    }
    if (upper) {
        for (int i = 1; i < n; i++) a[i] += a[0];
    }
}

template <typename T>
void divisors_mobius_transform(vector<T> &a, bool upper) {
    int n = a.size();
    vector<bool> is_prime(n, true);
    if (upper) {
        for (int i = 1; i < n; i++) a[i] -= a[0];
    }
    for (int i = 2; i < n; i++) {
        if (!is_prime[i]) continue;
        if (upper) {
            for (int j = 1; j * i < n; j++) {
                is_prime[j * i] = false;
                a[j] -= a[j * i];
            }
        } else {
            for (int j = (n - 1) / i; j > 0; j--) {
                is_prime[j * i] = false;
                a[j * i] -= a[j];
            }
        }
    }
    if (!upper) {
        for (int i = 1; i < n; i++) a[0] -= a[i];
    }
}

void solve() {}

int main() {
    int N, M;
    cin >> N >> M;

    vector<int> cnt(M + 1, 0);
    rep(i, N) {
        int x;
        cin >> x;
        cnt[x]++;
    }

    vector<mint> a(M + 1, 0);
    rep2(i, 1, M + 1) {
        int c = 0;
        for (int j = i; j <= M; j += i) c += cnt[j];
        a[i] = mint(2).pow(c) - 1;
    }

    divisors_mobius_transform(a, true);

    rep2(i, 1, M + 1) cout << a[i] << '\n';
}