You are the organizer of the famous "Zurich Music Festival". There will be n singers who will perform at the festival, identified by the integers 1, 2, ..., n. You must choose in which order they are going to perform on stage. 

You have m friends and each of them has a set of favourite singers. More precisely, for each 1≤ i≤ m, the i-th friend likes singers s_{i,1},   s_{i, 2},   ..., \,s_{i, q_i}.

A friend of yours is happy if the singers he likes perform consecutively (in an arbitrary order). An ordering of the singers is valid if it makes all your friends happy.

Compute the number of valid orderings modulo 998 244 353.

Input

The first line contains two integers n and m (1≤ n,\,m≤ 100) — the number of singers and the number of friends correspondingly.

The i-th of the next m lines contains the integer q_i (1≤ q_i≤ n) — the number of favorite singers of the i-th friend – followed by the q_i integers s_{i,1},   s_{i, 2},   ..., \,s_{i, q_i} (1≤ s_{i,1}<s_{i,2}<⋅⋅⋅<s_{i,q_i}≤ n) — the indexes of his favorite singers.

Output

Print the number of valid orderings of the singers modulo 998 244 353.

Examples

Input


3 1
2 1 3


Output


4


Input


5 5
2 1 2
2 2 3
2 3 4
2 4 5
2 1 5


Output


0


Input


100 1
1 50


Output


35305197


Input


5 1
5 1 2 3 4 5


Output


120


Input


2 5
1 2
1 2
1 2
1 1
1 1


Output


2


Input


11 4
5 4 5 7 9 11
2 2 10
2 9 11
7 1 2 3 5 8 10 11


Output


384

Note

Explanation of the first sample: There are 3 singers and only 1 friend. The friend likes the two singers 1 and 3. Thus, the 4 valid orderings are: 

  * 1 3 2
  * 2 1 3
  * 2 3 1
  * 3 1 2



Explanation of the second sample: There are 5 singers and 5 friends. One can show that no ordering is valid.

Explanation of the third sample: There are 100 singers and only 1 friend. The friend likes only singer 50, hence all the 100! possible orderings are valid.

Explanation of the fourth sample: There are 5 singers and only 1 friend. The friend likes all the singers, hence all the 5!=120 possible orderings are valid.
#include <bits/stdc++.h>
using namespace std;
 
using ll = long long;
using db = long double; // or double, if TL is tight
using str = string; // yay python!

using pi = pair<int,int>;
using pl = pair<ll,ll>;
using pd = pair<db,db>;

using vi = vector<int>;
using vb = vector<bool>;
using vl = vector<ll>;
using vd = vector<db>; 
using vs = vector<str>;
using vpi = vector<pi>;
using vpl = vector<pl>; 
using vpd = vector<pd>;

#define tcT template<class T
#define tcTU tcT, class U
// ^ lol this makes everything look weird but I'll try it
tcT> using V = vector<T>; 
tcT, size_t SZ> using AR = array<T,SZ>; 
tcT> using PR = pair<T,T>;

// pairs
#define mp make_pair
#define f first
#define s second

// vectors
// oops size(x), rbegin(x), rend(x) need C++17
#define sz(x) int((x).size())
#define bg(x) begin(x)
#define all(x) bg(x), end(x)
#define rall(x) x.rbegin(), x.rend() 
#define sor(x) sort(all(x)) 
#define rsz resize
#define ins insert 
#define ft front()
#define bk back()
#define pb push_back
#define eb emplace_back 
#define pf push_front
#define rtn return

#define lb lower_bound
#define ub upper_bound 
tcT> int lwb(V<T>& a, const T& b) { return int(lb(all(a),b)-bg(a)); }

// loops
#define FOR(i,a,b) for (int i = (a); i < (b); ++i)
#define F0R(i,a) FOR(i,0,a)
#define ROF(i,a,b) for (int i = (b)-1; i >= (a); --i)
#define R0F(i,a) ROF(i,0,a)
#define rep(a) F0R(_,a)
#define each(a,x) for (auto& a: x)

const int MOD = 998244353;
const int MX = 2e5+5;
const ll INF = 1e18; // not too close to LLONG_MAX
const db PI = acos((db)-1);
const int dx[4] = {1,0,-1,0}, dy[4] = {0,1,0,-1}; // for every grid problem!!
mt19937 rng((uint32_t)chrono::steady_clock::now().time_since_epoch().count()); 
template<class T> using pqg = priority_queue<T,vector<T>,greater<T>>;

// bitwise ops
// also see https://gcc.gnu.org/onlinedocs/gcc/Other-Builtins.html
constexpr int pct(int x) { return __builtin_popcount(x); } // # of bits set
constexpr int bits(int x) { // assert(x >= 0); // make C++11 compatible until USACO updates ...
    return x == 0 ? 0 : 31-__builtin_clz(x); } // floor(log2(x)) 
constexpr int p2(int x) { return 1<<x; }
constexpr int msk2(int x) { return p2(x)-1; }

ll cdiv(ll a, ll b) { return a/b+((a^b)>0&&a%b); } // divide a by b rounded up
ll fdiv(ll a, ll b) { return a/b-((a^b)<0&&a%b); } // divide a by b rounded down

tcT> bool ckmin(T& a, const T& b) {
    return b < a ? a = b, 1 : 0; } // set a = min(a,b)
tcT> bool ckmax(T& a, const T& b) {
    return a < b ? a = b, 1 : 0; }

tcTU> T fstTrue(T lo, T hi, U f) {
    hi ++; assert(lo <= hi); // assuming f is increasing
    while (lo < hi) { // find first index such that f is true 
        T mid = lo+(hi-lo)/2;
        f(mid) ? hi = mid : lo = mid+1; 
    } 
    return lo;
}
tcTU> T lstTrue(T lo, T hi, U f) {
    lo --; assert(lo <= hi); // assuming f is decreasing
    while (lo < hi) { // find first index such that f is true 
        T mid = lo+(hi-lo+1)/2;
        f(mid) ? lo = mid : hi = mid-1;
    } 
    return lo;
}
tcT> void remDup(vector<T>& v) { // sort and remove duplicates
    sort(all(v)); v.erase(unique(all(v)),end(v)); }
tcTU> void erase(T& t, const U& u) { // don't erase
    auto it = t.find(u); assert(it != end(t));
    t.erase(it); } // element that doesn't exist from (multi)set

#define tcTUU tcT, class ...U

inline namespace Helpers {
    //////////// is_iterable
    // https://stackoverflow.com/questions/13830158/check-if-a-variable-type-is-iterable
    // this gets used only when we can call begin() and end() on that type
    tcT, class = void> struct is_iterable : false_type {};
    tcT> struct is_iterable<T, void_t<decltype(begin(declval<T>())),
                                      decltype(end(declval<T>()))
                                     >
                           > : true_type {};
    tcT> constexpr bool is_iterable_v = is_iterable<T>::value;

    //////////// is_readable
    tcT, class = void> struct is_readable : false_type {};
    tcT> struct is_readable<T,
            typename std::enable_if_t<
                is_same_v<decltype(cin >> declval<T&>()), istream&>
            >
        > : true_type {};
    tcT> constexpr bool is_readable_v = is_readable<T>::value;

    //////////// is_printable
    // // https://nafe.es/posts/2020-02-29-is-printable/
    tcT, class = void> struct is_printable : false_type {};
    tcT> struct is_printable<T,
            typename std::enable_if_t<
                is_same_v<decltype(cout << declval<T>()), ostream&>
            >
        > : true_type {};
    tcT> constexpr bool is_printable_v = is_printable<T>::value;
}

inline namespace Input {
    tcT> constexpr bool needs_input_v = !is_readable_v<T> && is_iterable_v<T>;
    tcTUU> void re(T& t, U&... u);
    tcTU> void re(pair<T,U>& p); // pairs

    // re: read
    tcT> typename enable_if<is_readable_v<T>,void>::type re(T& x) { cin >> x; } // default
    tcT> void re(complex<T>& c) { T a,b; re(a,b); c = {a,b}; } // complex
    tcT> typename enable_if<needs_input_v<T>,void>::type re(T& i); // ex. vectors, arrays
    tcTU> void re(pair<T,U>& p) { re(p.f,p.s); }
    tcT> typename enable_if<needs_input_v<T>,void>::type re(T& i) {
        each(x,i) re(x); }
    tcTUU> void re(T& t, U&... u) { re(t); re(u...); } // read multiple

    // rv: resize and read vectors
    void rv(size_t) {}
    tcTUU> void rv(size_t N, V<T>& t, U&... u);
    template<class...U> void rv(size_t, size_t N2, U&... u);
    tcTUU> void rv(size_t N, V<T>& t, U&... u) {
        t.rsz(N); re(t);
        rv(N,u...); }
    template<class...U> void rv(size_t, size_t N2, U&... u) {
        rv(N2,u...); }

    // dumb shortcuts to read in ints
    void decrement() {} // subtract one from each
    tcTUU> void decrement(T& t, U&... u) { --t; decrement(u...); }
    #define ints(...) int __VA_ARGS__; re(__VA_ARGS__);
    #define int1(...) ints(__VA_ARGS__); decrement(__VA_ARGS__);
}

inline namespace ToString {
    tcT> constexpr bool needs_output_v = !is_printable_v<T> && is_iterable_v<T>;

    // ts: string representation to print
    tcT> typename enable_if<is_printable_v<T>,str>::type ts(T v) {
        stringstream ss; ss << fixed << setprecision(15) << v;
        return ss.str(); } // default
    tcT> str bit_vec(T t) { // bit vector to string
        str res = "{"; F0R(i,sz(t)) res += ts(t[i]);
        res += "}"; return res; }
    str ts(V<bool> v) { return bit_vec(v); }
    template<size_t SZ> str ts(bitset<SZ> b) { return bit_vec(b); } // bit vector
    tcTU> str ts(pair<T,U> p); // pairs
    tcT> typename enable_if<needs_output_v<T>,str>::type ts(T v); // vectors, arrays
    tcTU> str ts(pair<T,U> p) { return "("+ts(p.f)+", "+ts(p.s)+")"; }
    tcT> typename enable_if<is_iterable_v<T>,str>::type ts_sep(T v, str sep) {
        // convert container to string w/ separator sep
        bool fst = 1; str res = "";
        for (const auto& x: v) {
            if (!fst) res += sep;
            fst = 0; res += ts(x);
        }
        return res;
    }
    tcT> typename enable_if<needs_output_v<T>,str>::type ts(T v) {
        return "{"+ts_sep(v,", ")+"}"; }

    // for nested DS
    template<int, class T> typename enable_if<!needs_output_v<T>,vs>::type 
      ts_lev(const T& v) { return {ts(v)}; }
    template<int lev, class T> typename enable_if<needs_output_v<T>,vs>::type 
      ts_lev(const T& v) {
        if (lev == 0 || !sz(v)) return {ts(v)};
        vs res;
        for (const auto& t: v) {
            if (sz(res)) res.bk += ",";
            vs tmp = ts_lev<lev-1>(t);
            res.ins(end(res),all(tmp));
        }
        F0R(i,sz(res)) {
            str bef = " "; if (i == 0) bef = "{";
            res[i] = bef+res[i];
        }
        res.bk += "}";
        return res;
    }
}

inline namespace Output {
    template<class T> void pr_sep(ostream& os, str, const T& t) { os << ts(t); }
    template<class T, class... U> void pr_sep(ostream& os, str sep, const T& t, const U&... u) {
        pr_sep(os,sep,t); os << sep; pr_sep(os,sep,u...); }
    // print w/ no spaces
    template<class ...T> void pr(const T&... t) { pr_sep(cout,"",t...); } 
    // print w/ spaces, end with newline
    void ps() { cout << "\n"; }
    template<class ...T> void ps(const T&... t) { pr_sep(cout," ",t...); ps(); } 
    // debug to cerr
    template<class ...T> void dbg_out(const T&... t) {
        pr_sep(cerr," | ",t...); cerr << endl; }
    void loc_info(int line, str names) {
        cerr << "Line(" << line << ") -> [" << names << "]: "; }
    template<int lev, class T> void dbgl_out(const T& t) {
        cerr << "\n\n" << ts_sep(ts_lev<lev>(t),"\n") << "\n" << endl; }
    #ifdef LOCAL
        #define dbg(...) loc_info(__LINE__,#__VA_ARGS__), dbg_out(__VA_ARGS__)
        #define dbgl(lev,x) loc_info(__LINE__,#x), dbgl_out<lev>(x)
    #else // don't actually submit with this
        #define dbg(...) 0
        #define dbgl(lev,x) 0
    #endif
}

inline namespace FileIO {
    void setIn(str s)  { freopen(s.c_str(),"r",stdin); }
    void setOut(str s) { freopen(s.c_str(),"w",stdout); }
    void setIO(str s = "") {
        cin.tie(0)->sync_with_stdio(0); // unsync C / C++ I/O streams
        // cin.exceptions(cin.failbit);
        // throws exception when do smth illegal
        // ex. try to read letter into int
        if (sz(s)) setIn(s+".in"), setOut(s+".out"); // for old USACO
    }
}

const int mx = 105;
/**
 * Description: Disjoint Set Union with path compression
     * and union by size. Add edges and test connectivity. 
     * Use for Kruskal's or Boruvka's minimum spanning tree.
 * Time: O(\alpha(N))
 * Source: CSAcademy, KACTL
 * Verification: *
 */

struct DSU {
    vi e; void init(int N) { e = vi(N,-1); }
    int get(int x) { return e[x] < 0 ? x : e[x] = get(e[x]); } 
    bool sameSet(int a, int b) { return get(a) == get(b); }
    int size(int x) { return -e[get(x)]; }
    bool unite(int x, int y) { // union by size
        x = get(x), y = get(y); if (x == y) return 0;
        if (e[x] > e[y]) swap(x,y);
        e[x] += e[y]; e[y] = x; return 1;
    }
};

/**template<class T> T kruskal(int N, vector<pair<T,pi>> ed) {
    sort(all(ed));
    T ans = 0; DSU D; D.init(N); // edges that unite are in MST
    each(a,ed) if (D.unite(a.s.f,a.s.s)) ans += a.f; 
    return ans;
}*/

/**
 * Description: modular arithmetic operations 
 * Source: 
    * KACTL
    * https://codeforces.com/blog/entry/63903
    * https://codeforces.com/contest/1261/submission/65632855 (tourist)
    * https://codeforces.com/contest/1264/submission/66344993 (ksun)
    * also see https://github.com/ecnerwala/cp-book/blob/master/src/modnum.hpp (ecnerwal)
 * Verification: 
    * https://open.kattis.com/problems/modulararithmetic
 */

template<int MOD, int RT> struct mint {
    static const int mod = MOD;
    static constexpr mint rt() { return RT; } // primitive root for FFT
    int v; explicit operator int() const { return v; } // explicit -> don't silently convert to int
    mint() { v = 0; }
    mint(ll _v) { v = int((-MOD < _v && _v < MOD) ? _v : _v % MOD);
        if (v < 0) v += MOD; }
    friend bool operator==(const mint& a, const mint& b) { 
        return a.v == b.v; }
    friend bool operator!=(const mint& a, const mint& b) { 
        return !(a == b); }
    friend bool operator<(const mint& a, const mint& b) { 
        return a.v < b.v; }
    friend void re(mint& a) { ll x; re(x); a = mint(x); }
    friend str ts(mint a) { return ts(a.v); }
   
    mint& operator+=(const mint& m) { 
        if ((v += m.v) >= MOD) v -= MOD; 
        return *this; }
    mint& operator-=(const mint& m) { 
        if ((v -= m.v) < 0) v += MOD; 
        return *this; }
    mint& operator*=(const mint& m) { 
        v = int((ll)v*m.v%MOD); return *this; }
    mint& operator/=(const mint& m) { return (*this) *= inv(m); }
    friend mint pow(mint a, ll p) {
        mint ans = 1; assert(p >= 0);
        for (; p; p /= 2, a *= a) if (p&1) ans *= a;
        return ans; }
    friend mint inv(const mint& a) { assert(a.v != 0); 
        return pow(a,MOD-2); }
        
    mint operator-() const { return mint(-v); }
    mint& operator++() { return *this += 1; }
    mint& operator--() { return *this -= 1; }
    friend mint operator+(mint a, const mint& b) { return a += b; }
    friend mint operator-(mint a, const mint& b) { return a -= b; }
    friend mint operator*(mint a, const mint& b) { return a *= b; }
    friend mint operator/(mint a, const mint& b) { return a /= b; }
};

typedef mint<MOD,5> mi; // 5 is primitive root for both common mods
typedef vector<mi> vmi;
typedef pair<mi,mi> pmi;
typedef vector<pmi> vpmi;

vector<vmi> scmb; // small combinations
void genComb(int SZ) {
    scmb.assign(SZ,vmi(SZ)); scmb[0][0] = 1;
    FOR(i,1,SZ) F0R(j,i+1) 
        scmb[i][j] = scmb[i-1][j]+(j?scmb[i-1][j-1]:0);
}

/**
 * Description: pre-compute factorial mod inverses,
     * assumes $MOD$ is prime and $SZ < MOD$.
 * Time: O(SZ)
 * Source: KACTL
 * Verification: https://dmoj.ca/problem/tle17c4p5
 */

vmi invs, fac, ifac; // make sure to convert to LL before doing any multiplications ...
void genFac(int SZ) {
    invs.rsz(SZ), fac.rsz(SZ), ifac.rsz(SZ); 
    invs[1] = fac[0] = ifac[0] = 1; 
    FOR(i,2,SZ) invs[i] = mi(-(ll)MOD/i*(int)invs[MOD%i]);
    FOR(i,1,SZ) {
        fac[i] = fac[i-1]*i;
        ifac[i] = ifac[i-1]*invs[i];
    }
}
mi comb(int a, int b) {
    if (a < b || b < 0) return 0;
    return fac[a]*ifac[b]*ifac[a-b];
}

typedef bitset<mx> B;

const B B_ZERO;
B B_FULL;

void bitsetRemDup(vector<B>& v){
    vector<B> new_v;
    for(auto b: v){
        bool already_in = 0;
        for(auto in: new_v){
            if(b == in) already_in = 1;
        }

        if(!already_in) new_v.pb(b);
    }

    v = new_v;
}

int n, m;
vector<B> friends;

DSU dsu;
vi comp_adj[mx];
vi comp_nodes[mx];
// vpi comp_edges[mx];
// vpi comp_tree_edges[mx];
B comp_P[mx];
int par[mx]; //tree parent
vi children[mx]; //tree children

bool isAncestor(int C, int D){ // is C ancestor of D
    assert(C != D);
    if((comp_P[D]&comp_P[C]) == comp_P[D]){
        if(comp_P[D] == comp_P[C]){
            if(sz(comp_nodes[C]) == 1) return true;
            return false;
        }
        else{
            return true;
        }
    }
    return false;
}

bool visited[mx];
vi order;
void buildOrder(int node){
    visited[node] = 1;
    order.pb(node);
    for(auto u: comp_adj[node]){
        if(visited[u]) continue;
        buildOrder(u);
    }
}

bool isContiguous(vi a){
    assert(sz(a));
    for(int i = 0; i+1 < sz(a); i++){
        if(a[i]+1 != a[i+1]) return false;
    }
    return true;
}

int main() {
    setIO();
    #warning change mx back to 105
    genFac(100005);
    cin >> n >> m;

    for(int i = 1; i <= n; i++){
        B_FULL[i] = 1;
    }

    for(int i = 1; i <= m; i++){
        B b;
        int qi;
        cin >> qi;
        for(int j = 1; j <= qi; j++){
            int s;
            cin >> s;
            b[s] = 1;
        }
        friends.pb(b);
    }
    B full; for(int i = 1; i <= n; i++) full[i] = 1;
    friends.pb(full);
    bitsetRemDup(friends);
    m = sz(friends);

    dbg(m, friends);

    dsu.init(m);
    vpi edges;
    vpi tree_edges;
    for(int i = 0; i < m; i++){
        for(int j = i+1; j < m; j++){
            B isect = friends[i]&friends[j];
            if(isect != B_ZERO && isect != friends[i] && isect != friends[j]){
                //add edge i-j
                edges.pb(mp(i, j));
                if(dsu.unite(i, j)){
                    tree_edges.pb(mp(i, j));
                }
            }
        }
    }

    for(auto u: edges){
        comp_adj[u.f].pb(u.s);
        comp_adj[u.s].pb(u.f);
    }
    // for(auto u: edges){
    //     comp_edges[dsu.get(u.f)].pb(u);
    // }
    // for(auto u: tree_edges){
    //     comp_tree_edges[dsu.get(u.f)].pb(u);
    // }

    for(int i = 0; i < m; i++){
        comp_nodes[dsu.get(i)].pb(i);
        comp_P[dsu.get(i)]|=friends[i];
    }

    for(int i = 0; i < m; i++){
        dbg(i, comp_nodes[i], comp_P[i]);
    }

    int R = -1;
    for(int i = 0; i < m; i++){
        if(dsu.get(i) != i) continue;
        //get parent of this component
        par[i] = -1;
        for(int j = 0; j < m; j++){
            if(dsu.get(j) != j) continue;
            if(j == i) continue;
            if(!isAncestor(j, i)) continue;
            if(par[i] != -1 && isAncestor(j, par[i])) continue;

            par[i] = j;
        }

        if(par[i] == -1){
            assert(R == -1);
            R = i;
        }
    }

    assert(R != -1);

    for(int i = 0; i < m; i++){
        if(dsu.get(i) != i) continue;
        if(i == R) continue;
        children[par[i]].pb(i);
    }

    mi ans = mi(1);

    for(int i = 0; i < m; i++){
        if(dsu.get(i) != i) continue;
        dbg("COMPONENT", i);
        order.clear();
        buildOrder(i);

        assert(sz(order));

        vector<B> Z(2); //0 index is meaningless
        Z[1] = friends[order[0]];

        dbg(order);
        for(int j = 1; j < sz(order); j++){
            int k = sz(Z)-1;
            B S = friends[order[j]];
            vi I;
            vi nonempty_isect;

            B Z0 = S;
            for(int w = 1; w <= k; w++){
                Z0 = (Z0&(B_FULL^Z[w]));
            }

            dbg(Z, S);
            dbg(Z0);


            for(int w = 1; w <= k; w++){
                B Z_isect_S = Z[w]&S;
                B Z_minus_S = Z[w]^Z_isect_S;
                if(Z_isect_S != B_ZERO){
                    nonempty_isect.pb(w);
                    if(Z_minus_S != B_ZERO){
                        I.pb(w);
                    }
                }
            }
            assert(sz(nonempty_isect));

            dbg(I);
            dbg(nonempty_isect);

            if(!isContiguous(nonempty_isect)){
                ps(0);
                dbg("CASE1");
                exit(0);
            }

            if(sz(I) == 0){
                //prefix or suffix
                if(Z0 != B_ZERO){
                    if(nonempty_isect[0] == 1){ //prefix
                        assert(nonempty_isect.bk != k);
                        vector<B> new_Z(1);
                        new_Z.pb(Z0);
                        for(int i = 1; i <= k; i++){
                            new_Z.pb(Z[i]);
                        }
                        Z = new_Z;
                    }
                    else if(nonempty_isect.bk == k){ //suffix
                        Z.pb(Z0);
                    }
                    else{
                        ps(0);
                        dbg("CASE2");
                        exit(0);
                    }
                }
                else{
                    //everything is fine
                }
            }
            else if(sz(I) >= 3){
                ps(0);
                dbg("CASE3");
                exit(0);
            }
            else if(sz(I) == 2){
                if(Z0 != B_ZERO){
                    ps(0);
                    dbg("CASE4");
                    exit(0);
                }
                if(nonempty_isect[0] == I[0] && nonempty_isect.bk == I[1]){
                    int l = I[0];
                    int r = I[1];
                    B Zlp = Z[l]&S;
                    B Zlpp = Z[l]^Zlp;
                    B Zrp = Z[r]&S;
                    B Zrpp = Z[r]^Zrp;

                    vector<B> new_Z(1);
                    for(int i = 1; i <= l-1; i++){
                        new_Z.pb(Z[i]);
                    }
                    dbg(new_Z);
                    new_Z.pb(Zlpp); new_Z.pb(Zlp);
                    dbg(new_Z);
                    for(int i = l+1; i <= r-1; i++){
                        new_Z.pb(Z[i]);
                    }
                    dbg(new_Z);
                    new_Z.pb(Zrp); new_Z.pb(Zrpp);
                    dbg(new_Z);
                    for(int i = r+1; i <= k; i++){
                        new_Z.pb(Z[i]);
                    }
                    Z = new_Z;

                    dbg(new_Z, l, r);
                }
                else{
                    ps(0);
                    dbg("CASE5");
                    exit(0);
                }
            }
            else if(sz(I) == 1){
                if(Z0 != B_ZERO){
                    int t = I[0];
                    if(nonempty_isect[1] == t){
                        vector<B> new_Z(1);
                        B Ztp = Z[t]&S;
                        B Ztpp = Z[t]^Ztp;

                        new_Z.pb(Z0);
                        for(int i = 1; i < t; i++){
                            new_Z.pb(Z[i]);
                        }
                        new_Z.pb(Ztp); new_Z.pb(Ztpp);
                        for(int i = t+1; i <= k; i++){
                            new_Z.pb(Z[i]);
                        }
                        Z = new_Z;
                        dbg("HI");
                    }
                    else if(nonempty_isect[0] == t){
                        vector<B> new_Z(1);
                        B Ztp = Z[t]&S;
                        B Ztpp = Z[t]^Ztp;

                        for(int i = 1; i < t; i++){
                            new_Z.pb(Z[i]);
                        }
                        new_Z.pb(Ztpp); new_Z.pb(Ztp);
                        for(int i = t+1; i <= k; i++){
                            new_Z.pb(Z[i]);
                        }
                        new_Z.pb(Z0);
                        Z = new_Z;
                    }
                    else{
                        ps(0);
                        dbg("CASE6");
                        exit(0);
                    }
                }
                else{
                    int t = I[0];
                    B Ztp = Z[t]&S;
                    B Ztpp = Z[t]^Ztp;
                    assert(nonempty_isect[0] < t || nonempty_isect.bk > t);

                    if(nonempty_isect[0] < t){
                        assert(nonempty_isect.bk == t);

                        vector<B> new_Z(1);

                        for(int i = 1; i < t; i++){
                            new_Z.pb(Z[i]);
                        }
                        new_Z.pb(Ztp); new_Z.pb(Ztpp);
                        for(int i = t+1; i <= k; i++){
                            new_Z.pb(Z[i]);
                        }
                        Z = new_Z;
                    }
                    else if(nonempty_isect.bk > t){
                        vector<B> new_Z(1);

                        for(int i = 1; i < t; i++){
                            new_Z.pb(Z[i]);
                        }
                        new_Z.pb(Ztpp); new_Z.pb(Ztp);
                        for(int i = t+1; i <= k; i++){
                            new_Z.pb(Z[i]);
                        }
                        Z = new_Z;
                    }
                }
            }
        }

        dbg(Z);

        //multiply by 1 or 2
        if(sz(order) == 1){
            ans*=mi(1);
        }
        else{
            ans*=mi(2);
        }

        //mult by factorials
        for(int w = 1; w < sz(Z); w++){
            B E = Z[w];
            int fact_num = E.count();
            for(auto D: children[i]){
                if((comp_P[D]&E) == comp_P[D]){
                    fact_num-=(comp_P[D].count()-1);
                }
            }
            assert(fact_num >= 0);
            dbg(fact_num);
            ans*=fac[fact_num];
        }
    }

    ps(ans);

    // you should actually read the stuff at the bottom
}

/* stuff you should look for
    * int overflow, array bounds
    * special cases (n=1?)
    * do smth instead of nothing and stay organized
    * WRITE STUFF DOWN
    * DON'T GET STUCK ON ONE APPROACH
*/
