#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <chrono>
#include <cmath>
#include <complex>
#include <deque>
#include <forward_list>
#include <fstream>
#include <functional>
#include <iomanip>
#include <ios>
#include <iostream>
#include <limits>
#include <list>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <string>
#include <tuple>
#include <type_traits>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
using namespace std;
using lint = long long;
using pint = pair<int, int>;
using plint = pair<lint, lint>;
struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_;
#define ALL(x) (x).begin(), (x).end()
#define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i<i##_end_;i++)
#define IFOR(i, begin, end) for(int i=(end)-1,i##_begin_=(begin);i>=i##_begin_;i--)
#define REP(i, n) FOR(i,0,n)
#define IREP(i, n) IFOR(i,0,n)
template <typename T, typename V>
void ndarray(vector<T>& vec, const V& val, int len) { vec.assign(len, val); }
template <typename T, typename V, typename... Args> void ndarray(vector<T>& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); }
template <typename T> bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; }
template <typename T> bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; }
const std::vector<std::pair<int, int>> grid_dxs{{1, 0}, {-1, 0}, {0, 1}, {0, -1}};
int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); }
template <class T1, class T2> T1 floor_div(T1 num, T2 den) { return (num > 0 ? num / den : -((-num + den - 1) / den)); }
template <class T1, class T2> std::pair<T1, T2> operator+(const std::pair<T1, T2> &l, const std::pair<T1, T2> &r) { return std::make_pair(l.first + r.first, l.second + r.second); }
template <class T1, class T2> std::pair<T1, T2> operator-(const std::pair<T1, T2> &l, const std::pair<T1, T2> &r) { return std::make_pair(l.first - r.first, l.second - r.second); }
template <class T> std::vector<T> sort_unique(std::vector<T> vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; }
template <class T> int arglb(const std::vector<T> &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); }
template <class T> int argub(const std::vector<T> &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); }
template <class IStream, class T> IStream &operator>>(IStream &is, std::vector<T> &vec) { for (auto &v : vec) is >> v; return is; }

template <class OStream, class T> OStream &operator<<(OStream &os, const std::vector<T> &vec);
template <class OStream, class T, size_t sz> OStream &operator<<(OStream &os, const std::array<T, sz> &arr);
template <class OStream, class T, class TH> OStream &operator<<(OStream &os, const std::unordered_set<T, TH> &vec);
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const pair<T, U> &pa);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::deque<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::set<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::multiset<T> &vec);
template <class OStream, class T> OStream &operator<<(OStream &os, const std::unordered_multiset<T> &vec);
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const std::pair<T, U> &pa);
template <class OStream, class TK, class TV> OStream &operator<<(OStream &os, const std::map<TK, TV> &mp);
template <class OStream, class TK, class TV, class TH> OStream &operator<<(OStream &os, const std::unordered_map<TK, TV, TH> &mp);
template <class OStream, class... T> OStream &operator<<(OStream &os, const std::tuple<T...> &tpl);

template <class OStream, class T> OStream &operator<<(OStream &os, const std::vector<T> &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <class OStream, class T, size_t sz> OStream &operator<<(OStream &os, const std::array<T, sz> &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; }
template <class... T> std::istream &operator>>(std::istream &is, std::tuple<T...> &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; }
template <class OStream, class... T> OStream &operator<<(OStream &os, const std::tuple<T...> &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; }
template <class OStream, class T, class TH> OStream &operator<<(OStream &os, const std::unordered_set<T, TH> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::deque<T> &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::set<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T> OStream &operator<<(OStream &os, const std::unordered_multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <class OStream, class T, class U> OStream &operator<<(OStream &os, const std::pair<T, U> &pa) { return os << '(' << pa.first << ',' << pa.second << ')'; }
template <class OStream, class TK, class TV> OStream &operator<<(OStream &os, const std::map<TK, TV> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
template <class OStream, class TK, class TV, class TH> OStream &operator<<(OStream &os, const std::unordered_map<TK, TV, TH> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
#ifdef HITONANODE_LOCAL
const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m";
#define dbg(x) std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl
#define dbgif(cond, x) ((cond) ? std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl : std::cerr)
#else
#define dbg(x) ((void)0)
#define dbgif(cond, x) ((void)0)
#endif

// Subset sum (fast zeta transform)
// Complexity: O(N 2^N) for array of size 2^N
template <typename T> void subset_sum(std::vector<T> &f) {
    const int sz = f.size(), n = __builtin_ctz(sz);
    assert(__builtin_popcount(sz) == 1);
    for (int d = 0; d < n; d++) {
        for (int S = 0; S < 1 << n; S++)
            if (S & (1 << d)) f[S] += f[S ^ (1 << d)];
    }
}
// Inverse of subset sum (fast moebius transform)
// Complexity: O(N 2^N) for array of size 2^N
template <typename T> void subset_sum_inv(std::vector<T> &g) {
    const int sz = g.size(), n = __builtin_ctz(sz);
    assert(__builtin_popcount(sz) == 1);
    for (int d = 0; d < n; d++) {
        for (int S = 0; S < 1 << n; S++)
            if (S & (1 << d)) g[S] -= g[S ^ (1 << d)];
    }
}

// Superset sum / its inverse (fast zeta/moebius transform)
// Complexity: O(N 2^N) for array of size 2^N
template <typename T> void superset_sum(std::vector<T> &f) {
    const int sz = f.size(), n = __builtin_ctz(sz);
    assert(__builtin_popcount(sz) == 1);
    for (int d = 0; d < n; d++) {
        for (int S = 0; S < 1 << n; S++)
            if (!(S & (1 << d))) f[S] += f[S | (1 << d)];
    }
}
template <typename T> void superset_sum_inv(std::vector<T> &g) {
    const int sz = g.size(), n = __builtin_ctz(sz);
    assert(__builtin_popcount(sz) == 1);
    for (int d = 0; d < n; d++) {
        for (int S = 0; S < 1 << n; S++)
            if (!(S & (1 << d))) g[S] -= g[S | (1 << d)];
    }
}

template <typename T> std::vector<std::vector<T>> build_zeta_(int D, const std::vector<T> &f) {
    int n = f.size();
    std::vector<std::vector<T>> ret(D, std::vector<T>(n));
    for (int i = 0; i < n; i++) ret[__builtin_popcount(i)][i] += f[i];
    for (auto &vec : ret) subset_sum(vec);
    return ret;
}

template <typename T>
std::vector<T> get_moebius_of_prod_(const std::vector<std::vector<T>> &mat1,
                                    const std::vector<std::vector<T>> &mat2) {
    int D = mat1.size(), n = mat1[0].size();
    std::vector<std::vector<int>> pc2i(D);
    for (int i = 0; i < n; i++) pc2i[__builtin_popcount(i)].push_back(i);
    std::vector<T> tmp, ret(mat1[0].size());
    for (int d = 0; d < D; d++) {
        tmp.assign(mat1[d].size(), 0);
        for (int e = 0; e <= d; e++) {
            for (int i = 0; i < int(tmp.size()); i++) tmp[i] += mat1[e][i] * mat2[d - e][i];
        }
        subset_sum_inv(tmp);
        for (auto i : pc2i[d]) ret[i] = tmp[i];
    }
    return ret;
};

// Subset convolution
// h[S] = \sum_T f[T] * g[S - T]
// Complexity: O(N^2 2^N) for arrays of size 2^N
template <typename T> std::vector<T> subset_convolution(std::vector<T> f, std::vector<T> g) {
    const int sz = f.size(), m = __builtin_ctz(sz) + 1;
    assert(__builtin_popcount(sz) == 1 and f.size() == g.size());
    auto ff = build_zeta_(m, f), fg = build_zeta_(m, g);
    return get_moebius_of_prod_(ff, fg);
}

// https://hos-lyric.hatenablog.com/entry/2021/01/14/201231
template <class T, class Function> void subset_func(std::vector<T> &f, const Function &func) {
    const int sz = f.size(), m = __builtin_ctz(sz) + 1;
    assert(__builtin_popcount(sz) == 1);

    auto ff = build_zeta_(m, f);

    std::vector<T> p(m);
    for (int i = 0; i < sz; i++) {
        for (int d = 0; d < m; d++) p[d] = ff[d][i];
        func(p);
        for (int d = 0; d < m; d++) ff[d][i] = p[d];
    }

    for (auto &vec : ff) subset_sum_inv(vec);
    for (int i = 0; i < sz; i++) f[i] = ff[__builtin_popcount(i)][i];
}

// log(f(x)) for f(x), f(0) == 1
// Requires inv()
template <class T> void poly_log(std::vector<T> &f) {
    assert(f.at(0) == T(1));
    static std::vector<T> invs{0};
    const int m = f.size();
    std::vector<T> finv(m);
    for (int d = 0; d < m; d++) {
        finv[d] = (d == 0);
        if (int(invs.size()) <= d) invs.push_back(T(d).inv());
        for (int e = 0; e < d; e++) finv[d] -= finv[e] * f[d - e];
    }
    std::vector<T> ret(m);
    for (int d = 1; d < m; d++) {
        for (int e = 0; d + e < m; e++) ret[d + e] += f[d] * d * finv[e] * invs[d + e];
    }
    f = ret;
}

// log(f(S)) for set function f(S), f(0) == 1
// Requires inv()
// Complexity: O(n^2 2^n)
// https://atcoder.jp/contests/abc213/tasks/abc213_g
template <class T> void subset_log(std::vector<T> &f) { subset_func(f, poly_log<T>); }

// exp(f(S)) for set function f(S), f(0) == 0
// Complexity: O(n^2 2^n)
// https://codeforces.com/blog/entry/92183
template <class T> void subset_exp(std::vector<T> &f) {
    const int sz = f.size(), m = __builtin_ctz(sz);
    assert(sz == 1 << m);
    assert(f.at(0) == 0);
    std::vector<T> ret{T(1)};
    ret.reserve(sz);
    for (int d = 0; d < m; d++) {
        auto c = subset_convolution({f.begin() + (1 << d), f.begin() + (1 << (d + 1))}, ret);
        ret.insert(ret.end(), c.begin(), c.end());
    }
    f = ret;
}

// sqrt(f(x)), f(x) == 1
// Requires inv of 2
// Compelxity: O(n^2)
template <class T> void poly_sqrt(std::vector<T> &f) {
    assert(f.at(0) == T(1));
    const int m = f.size();
    static const auto inv2 = T(2).inv();
    for (int d = 1; d < m; d++) {
        if (~(d & 1)) f[d] -= f[d / 2] * f[d / 2];
        f[d] *= inv2;
        for (int e = 1; e < d - e; e++) f[d] -= f[e] * f[d - e];
    }
}

// sqrt(f(S)) for set function f(S), f(0) == 1
// Requires inv()
// https://atcoder.jp/contests/xmascon20/tasks/xmascon20_h
template <class T> void subset_sqrt(std::vector<T> &f) { subset_func(f, poly_sqrt<T>); }

// exp(f(S)) for set function f(S), f(0) == 0
template <class T> void poly_exp(std::vector<T> &P) {
    const int m = P.size();
    assert(m and P[0] == 0);
    std::vector<T> Q(m), logQ(m), Qinv(m);
    Q[0] = Qinv[0] = T(1);
    static std::vector<T> invs{0};

    auto set_invlog = [&](int d) {
        Qinv[d] = 0;
        for (int e = 0; e < d; e++) Qinv[d] -= Qinv[e] * Q[d - e];
        while (d >= int(invs.size())) {
            int sz = invs.size();
            invs.push_back(T(sz).inv());
        }
        logQ[d] = 0;
        for (int e = 1; e <= d; e++) logQ[d] += Q[e] * e * Qinv[d - e];
        logQ[d] *= invs[d];
    };
    for (int d = 1; d < m; d++) {
        Q[d] += P[d] - logQ[d];
        set_invlog(d);
        assert(logQ[d] == P[d]);
        if (d + 1 < m) set_invlog(d + 1);
    }
    P = Q;
}

// f(S)^k for set function f(S)
// Requires inv()
template <class T> void subset_pow(std::vector<T> &f, long long k) {
    auto poly_pow = [&](std::vector<T> &f) {
        const int m = f.size();
        if (k == 0) f[0] = 1, std::fill(f.begin() + 1, f.end(), T(0));
        if (k <= 1) return;
        int nzero = 0;
        while (nzero < int(f.size()) and f[nzero] == T(0)) nzero++;
        int rem = std::max<long long>((long long)f.size() - nzero * k, 0LL);
        if (rem == 0) {
            std::fill(f.begin(), f.end(), 0);
            return;
        }
        f.erase(f.begin(), f.begin() + nzero);
        f.resize(rem);
        const T f0 = f.at(0), f0inv = f0.inv(), f0pow = f0.pow(k);
        for (auto &x : f) x *= f0inv;
        poly_log(f);
        for (auto &x : f) x *= k;
        poly_exp(f);
        for (auto &x : f) x *= f0pow;
        f.resize(rem, 0);
        f.insert(f.begin(), m - int(f.size()), T(0));
    };
    subset_func(f, poly_pow);
}


// Count perfect matchings of undirected graph (Hafnian of the matrix)
// Assumption: mat[i][j] == mat[j][i], mat[i][i] == 0
// Complexity: O(n^2 2^n)
// [1] A. Björklund, "Counting Perfect Matchings as Fast as Ryser,
//     Proc. of 23rd ACM-SIAM symposium on Discrete Algorithms, pp.914-921, 2012.
template <class T> T hafnian(const std::vector<std::vector<T>> &mat) {
    const int N = mat.size();
    if (N % 2) return 0;
    std::vector<std::vector<std::vector<T>>> B(N, std::vector<std::vector<T>>(N));
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) B[i][j] = std::vector<T>{mat[i][j]};
    }

    std::vector<int> pc(1 << (N / 2));
    std::vector<std::vector<int>> pc2i(N / 2 + 1);
    for (int i = 0; i < int(pc.size()); i++) {
        pc[i] = __builtin_popcount(i);
        pc2i[pc[i]].push_back(i);
    }

    std::vector<T> h{1};
    for (int i = 1; i < N / 2; i++) {
        int r1 = N - i * 2, r2 = r1 + 1;
        auto h_add = subset_convolution(h, B[r2][r1]);
        h.insert(h.end(), h_add.begin(), h_add.end());

        std::vector<std::vector<std::vector<T>>> B1(r1), B2(r1);
        for (int j = 0; j < r1; j++) {
            B1[j] = build_zeta_(i, B[r1][j]);
            B2[j] = build_zeta_(i, B[r2][j]);
        }

        for (int j = 0; j < r1; j++) {
            for (int k = 0; k < j; k++) {
                auto Sijk1 = get_moebius_of_prod_(B1[j], B2[k]);
                auto Sijk2 = get_moebius_of_prod_(B1[k], B2[j]);
                for (int s = 0; s < int(Sijk2.size()); s++) Sijk1[s] += Sijk2[s];
                B[j][k].insert(B[j][k].end(), Sijk1.begin(), Sijk1.end());
            }
        }
    }
    T ret = 0;
    for (int i = 0; i < int(h.size()); i++) ret += h[i] * B[1][0][h.size() - 1 - i];
    return ret;
}


#include <cassert>
#include <chrono>
#include <random>

// F_p, p = 2^61 - 1
// https://qiita.com/keymoon/items/11fac5627672a6d6a9f6
class ModIntMersenne61 {
    static const long long md = (1LL << 61) - 1;
    long long _v;

    inline unsigned hi() const noexcept { return _v >> 31; }
    inline unsigned lo() const noexcept { return _v & ((1LL << 31) - 1); }

public:
    static long long mod() { return md; }

    ModIntMersenne61() : _v(0) {}
    // 0 <= x < md * 2
    explicit ModIntMersenne61(long long x) : _v(x >= md ? x - md : x) {}

    long long val() const noexcept { return _v; }

    ModIntMersenne61 operator+(const ModIntMersenne61 &x) const {
        return ModIntMersenne61(_v + x._v);
    }

    ModIntMersenne61 operator-(const ModIntMersenne61 &x) const {
        return ModIntMersenne61(_v + md - x._v);
    }

    ModIntMersenne61 operator*(const ModIntMersenne61 &x) const {
        using ull = unsigned long long;

        ull uu = (ull)hi() * x.hi() * 2;
        ull ll = (ull)lo() * x.lo();
        ull lu = (ull)hi() * x.lo() + (ull)lo() * x.hi();

        ull sum = uu + ll + ((lu & ((1ULL << 30) - 1)) << 31) + (lu >> 30);
        ull reduced = (sum >> 61) + (sum & ull(md));
        return ModIntMersenne61(reduced);
    }

    ModIntMersenne61 pow(long long n) const {
        assert(n >= 0);
        ModIntMersenne61 ans(1), tmp = *this;
        while (n) {
            if (n & 1) ans *= tmp;
            tmp *= tmp, n >>= 1;
        }
        return ans;
    }

    ModIntMersenne61 inv() const { return pow(md - 2); }

    ModIntMersenne61 operator/(const ModIntMersenne61 &x) const { return *this * x.inv(); }

    ModIntMersenne61 operator-() const { return ModIntMersenne61(md - _v); }
    ModIntMersenne61 &operator+=(const ModIntMersenne61 &x) { return *this = *this + x; }
    ModIntMersenne61 &operator-=(const ModIntMersenne61 &x) { return *this = *this - x; }
    ModIntMersenne61 &operator*=(const ModIntMersenne61 &x) { return *this = *this * x; }
    ModIntMersenne61 &operator/=(const ModIntMersenne61 &x) { return *this = *this / x; }

    ModIntMersenne61 operator+(unsigned x) const { return ModIntMersenne61(this->_v + x); }

    bool operator==(const ModIntMersenne61 &x) const { return _v == x._v; }
    bool operator!=(const ModIntMersenne61 &x) const { return _v != x._v; }
    bool operator<(const ModIntMersenne61 &x) const { return _v < x._v; } // To use std::map

    template <class OStream> friend OStream &operator<<(OStream &os, const ModIntMersenne61 &x) {
        return os << x._v;
    }

    static ModIntMersenne61 randgen(bool force_update = false) {
        static ModIntMersenne61 b(0);
        if (b == ModIntMersenne61(0) or force_update) {
            std::mt19937 mt(std::chrono::steady_clock::now().time_since_epoch().count());
            std::uniform_int_distribution<long long> d(1, ModIntMersenne61::mod());
            b = ModIntMersenne61(d(mt));
        }
        return b;
    }
};


#include <algorithm>
#include <cassert>
#include <cmath>
#include <iterator>
#include <type_traits>
#include <utility>
#include <vector>

namespace matrix_ {
struct has_id_method_impl {
    template <class T_> static auto check(T_ *) -> decltype(T_::id(), std::true_type());
    template <class T_> static auto check(...) -> std::false_type;
};
template <class T_> struct has_id : decltype(has_id_method_impl::check<T_>(nullptr)) {};
} // namespace matrix_

template <typename T> struct matrix {
    int H, W;
    std::vector<T> elem;
    typename std::vector<T>::iterator operator[](int i) { return elem.begin() + i * W; }
    inline T &at(int i, int j) { return elem[i * W + j]; }
    inline T get(int i, int j) const { return elem[i * W + j]; }
    int height() const { return H; }
    int width() const { return W; }
    std::vector<std::vector<T>> vecvec() const {
        std::vector<std::vector<T>> ret(H);
        for (int i = 0; i < H; i++) {
            std::copy(elem.begin() + i * W, elem.begin() + (i + 1) * W, std::back_inserter(ret[i]));
        }
        return ret;
    }
    operator std::vector<std::vector<T>>() const { return vecvec(); }
    matrix() = default;
    matrix(int H, int W) : H(H), W(W), elem(H * W) {}
    matrix(const std::vector<std::vector<T>> &d) : H(d.size()), W(d.size() ? d[0].size() : 0) {
        for (auto &raw : d) std::copy(raw.begin(), raw.end(), std::back_inserter(elem));
    }

    template <typename T2, typename std::enable_if<matrix_::has_id<T2>::value>::type * = nullptr>
    static T2 _T_id() {
        return T2::id();
    }
    template <typename T2, typename std::enable_if<!matrix_::has_id<T2>::value>::type * = nullptr>
    static T2 _T_id() {
        return T2(1);
    }

    static matrix Identity(int N) {
        matrix ret(N, N);
        for (int i = 0; i < N; i++) ret.at(i, i) = _T_id<T>();
        return ret;
    }

    matrix operator-() const {
        matrix ret(H, W);
        for (int i = 0; i < H * W; i++) ret.elem[i] = -elem[i];
        return ret;
    }
    matrix operator*(const T &v) const {
        matrix ret = *this;
        for (auto &x : ret.elem) x *= v;
        return ret;
    }
    matrix operator/(const T &v) const {
        matrix ret = *this;
        const T vinv = _T_id<T>() / v;
        for (auto &x : ret.elem) x *= vinv;
        return ret;
    }
    matrix operator+(const matrix &r) const {
        matrix ret = *this;
        for (int i = 0; i < H * W; i++) ret.elem[i] += r.elem[i];
        return ret;
    }
    matrix operator-(const matrix &r) const {
        matrix ret = *this;
        for (int i = 0; i < H * W; i++) ret.elem[i] -= r.elem[i];
        return ret;
    }
    matrix operator*(const matrix &r) const {
        matrix ret(H, r.W);
        for (int i = 0; i < H; i++) {
            for (int k = 0; k < W; k++) {
                for (int j = 0; j < r.W; j++) ret.at(i, j) += this->get(i, k) * r.get(k, j);
            }
        }
        return ret;
    }
    matrix &operator*=(const T &v) { return *this = *this * v; }
    matrix &operator/=(const T &v) { return *this = *this / v; }
    matrix &operator+=(const matrix &r) { return *this = *this + r; }
    matrix &operator-=(const matrix &r) { return *this = *this - r; }
    matrix &operator*=(const matrix &r) { return *this = *this * r; }
    bool operator==(const matrix &r) const { return H == r.H and W == r.W and elem == r.elem; }
    bool operator!=(const matrix &r) const { return H != r.H or W != r.W or elem != r.elem; }
    bool operator<(const matrix &r) const { return elem < r.elem; }
    matrix pow(int64_t n) const {
        matrix ret = Identity(H);
        bool ret_is_id = true;
        if (n == 0) return ret;
        for (int i = 63 - __builtin_clzll(n); i >= 0; i--) {
            if (!ret_is_id) ret *= ret;
            if ((n >> i) & 1) ret *= (*this), ret_is_id = false;
        }
        return ret;
    }
    std::vector<T> pow_vec(int64_t n, std::vector<T> vec) const {
        matrix x = *this;
        while (n) {
            if (n & 1) vec = x * vec;
            x *= x;
            n >>= 1;
        }
        return vec;
    };
    matrix transpose() const {
        matrix ret(W, H);
        for (int i = 0; i < H; i++) {
            for (int j = 0; j < W; j++) ret.at(j, i) = this->get(i, j);
        }
        return ret;
    }
    // Gauss-Jordan elimination
    // - Require inverse for every non-zero element
    // - Complexity: O(H^2 W)
    template <typename T2, typename std::enable_if<std::is_floating_point<T2>::value>::type * = nullptr>
    static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept {
        int piv = -1;
        for (int j = h; j < mtr.H; j++) {
            if (mtr.get(j, c) and (piv < 0 or std::abs(mtr.get(j, c)) > std::abs(mtr.get(piv, c))))
                piv = j;
        }
        return piv;
    }
    template <typename T2, typename std::enable_if<!std::is_floating_point<T2>::value>::type * = nullptr>
    static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept {
        for (int j = h; j < mtr.H; j++) {
            if (mtr.get(j, c) != T2()) return j;
        }
        return -1;
    }
    matrix gauss_jordan() const {
        int c = 0;
        matrix mtr(*this);
        std::vector<int> ws;
        ws.reserve(W);
        for (int h = 0; h < H; h++) {
            if (c == W) break;
            int piv = choose_pivot(mtr, h, c);
            if (piv == -1) {
                c++;
                h--;
                continue;
            }
            if (h != piv) {
                for (int w = 0; w < W; w++) {
                    std::swap(mtr[piv][w], mtr[h][w]);
                    mtr.at(piv, w) *= -_T_id<T>(); // To preserve sign of determinant
                }
            }
            ws.clear();
            for (int w = c; w < W; w++) {
                if (mtr.at(h, w) != T()) ws.emplace_back(w);
            }
            const T hcinv = _T_id<T>() / mtr.at(h, c);
            for (int hh = 0; hh < H; hh++)
                if (hh != h) {
                    const T coeff = mtr.at(hh, c) * hcinv;
                    for (auto w : ws) mtr.at(hh, w) -= mtr.at(h, w) * coeff;
                    mtr.at(hh, c) = T();
                }
            c++;
        }
        return mtr;
    }
    int rank_of_gauss_jordan() const {
        for (int i = H * W - 1; i >= 0; i--) {
            if (elem[i] != 0) return i / W + 1;
        }
        return 0;
    }
    int rank() const { return gauss_jordan().rank_of_gauss_jordan(); }

    T determinant_of_upper_triangle() const {
        T ret = _T_id<T>();
        for (int i = 0; i < H; i++) ret *= get(i, i);
        return ret;
    }
    int inverse() {
        assert(H == W);
        std::vector<std::vector<T>> ret = Identity(H), tmp = *this;
        int rank = 0;
        for (int i = 0; i < H; i++) {
            int ti = i;
            while (ti < H and tmp[ti][i] == 0) ti++;
            if (ti == H) {
                continue;
            } else {
                rank++;
            }
            ret[i].swap(ret[ti]), tmp[i].swap(tmp[ti]);
            T inv = _T_id<T>() / tmp[i][i];
            for (int j = 0; j < W; j++) ret[i][j] *= inv;
            for (int j = i + 1; j < W; j++) tmp[i][j] *= inv;
            for (int h = 0; h < H; h++) {
                if (i == h) continue;
                const T c = -tmp[h][i];
                for (int j = 0; j < W; j++) ret[h][j] += ret[i][j] * c;
                for (int j = i + 1; j < W; j++) tmp[h][j] += tmp[i][j] * c;
            }
        }
        *this = ret;
        return rank;
    }
    friend std::vector<T> operator*(const matrix &m, const std::vector<T> &v) {
        assert(m.W == int(v.size()));
        std::vector<T> ret(m.H);
        for (int i = 0; i < m.H; i++) {
            for (int j = 0; j < m.W; j++) ret[i] += m.get(i, j) * v[j];
        }
        return ret;
    }
    friend std::vector<T> operator*(const std::vector<T> &v, const matrix &m) {
        assert(int(v.size()) == m.H);
        std::vector<T> ret(m.W);
        for (int i = 0; i < m.H; i++) {
            for (int j = 0; j < m.W; j++) ret[j] += v[i] * m.get(i, j);
        }
        return ret;
    }
    std::vector<T> prod(const std::vector<T> &v) const { return (*this) * v; }
    std::vector<T> prod_left(const std::vector<T> &v) const { return v * (*this); }
    template <class OStream> friend OStream &operator<<(OStream &os, const matrix &x) {
        os << "[(" << x.H << " * " << x.W << " matrix)";
        os << "\n[column sums: ";
        for (int j = 0; j < x.W; j++) {
            T s = T();
            for (int i = 0; i < x.H; i++) s += x.get(i, j);
            os << s << ",";
        }
        os << "]";
        for (int i = 0; i < x.H; i++) {
            os << "\n[";
            for (int j = 0; j < x.W; j++) os << x.get(i, j) << ",";
            os << "]";
        }
        os << "]\n";
        return os;
    }
    template <class IStream> friend IStream &operator>>(IStream &is, matrix &x) {
        for (auto &v : x.elem) is >> v;
        return is;
    }
};

using mint = ModIntMersenne61;
int main() {
    constexpr int D = 19;
    vector<vector<int>> vs(D);
    matrix<mint> mat(D, D);

    vector<lint> dp(1 << D);
    dp.front() = 1;

    REP(d, D) {
        vector<lint> dpnxt = dp;
        auto &v = vs.at(d);
        int l;
        cin >> l;
        v.resize(l);
        for (auto &x : v) {
            cin >> x;
            --x;
            REP(s, dp.size()) {
                if ((s & (1 << x)) == 0) {
                    dpnxt.at(s + (1 << x)) += dp.at(s);
                }
            }
            mat[d][x] = mint(1);
        }
        dp = dpnxt;
    }
    // dbg(vs);
    // dbg(mat);
    auto de = mat.gauss_jordan().determinant_of_upper_triangle();
    lint det = de.val();
    if (det > 1LL << 60) det = -(-de).val();
    dbg(det);
    dbg(dp.back());
    dbg(dp.back() / 2);
    cout << (dp.back() + det) / 2 << ' ' << (dp.back() - det) / 2 << endl;
}