//#pragma GCC optimize ("-O3") #include using namespace std; //@起動時 struct initon { initon() { cin.tie(0); ios::sync_with_stdio(false); cout.setf(ios::fixed); cout.precision(16); srand((unsigned) clock() + (unsigned) time(NULL)); }; } __initon; //衝突対策 #define ws ___ws struct T { int f, s, t; T() { f = -1, s = -1, t = -1; } T(int f, int s, int t) : f(f), s(s), t(t) {} bool operator<(const T &r) const { return f != r.f ? f < r.f : s != r.s ? s < r.s : t < r.t; //return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t; 大きい順 } bool operator>(const T &r) const { return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t; //return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t; 小さい順 } bool operator==(const T &r) const { return f == r.f && s == r.s && t == r.t; } bool operator!=(const T &r) const { return f != r.f || s != r.s || t != r.t; } int operator[](int i) { assert(i < 3); return i == 0 ? f : i == 1 ? s : t; } }; #define int long long #define ll long long //#define double long double #define pow powl #define ull unsigned long long using dou = double; using itn = int; using str = string; using bo= bool; #define au auto using P = pair; #define fi first #define se second #define vec vector #define beg begin #define rbeg rbegin #define con continue #define bre break #define brk break #define is == //マクロ省略系 コンテナ //マクロ省略系 コンテナ using vi = vector; using vb = vector; using vs = vector; using vd = vector; using vc = vector; using vp = vector

; using vt = vector; #define vec vector #define o_vvt(o1, o2, o3, o4, name, ...) name #define vvt0(t) vec> #define vvt1(t, a) vec>a #define vvt2(t, a, b) vec>a(b) #define vvt3(t, a, b, c) vec> a(b,vec(c)) #define vvt4(t, a, b, c, d) vec> a(b,vec(c,d)) #define vvi(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(ll,__VA_ARGS__) #define vvb(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(bool,__VA_ARGS__) #define vvs(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(string,__VA_ARGS__) #define vvd(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(double,__VA_ARGS__) #define vvc(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(char,__VA_ARGS__) #define vvp(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(P,__VA_ARGS__) #define vvt(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(T,__VA_ARGS__) template vector make_v(size_t a) { return vector(a); } template auto make_v(size_t a, Ts... ts) { return vector(ts...))>(a, make_v(ts...)); } #define vni(name, ...) auto name = make_v(__VA_ARGS__) #define vnb(name, ...) auto name = make_v(__VA_ARGS__) #define vns(name, ...) auto name = make_v(__VA_ARGS__) #define vnd(name, ...) auto name = make_v(__VA_ARGS__) #define vnc(name, ...) auto name = make_v(__VA_ARGS__) #define vnp(name, ...) auto name = make_v

(__VA_ARGS__) #define v3i(a, b, c, d) vector> a(b, vector(c, vi(d))) #define v3d(a, b, c, d) vector> a(b, vector(c, vd(d))) #define v3m(a, b, c, d) vector> a(b, vector(c, vm(d))) #define _vvi vector #define _vvl vector #define _vvb vector #define _vvs vector #define _vvd vector #define _vvc vector #define _vvp vector #define PQ priority_queue, greater > #define tos to_string using mapi = map; using mapd = map; using mapc = map; using maps = map; using seti = set; using setd = set; using setc = set; using sets = set; using qui = queue; #define bset bitset #define uset unordered_set #define mset multiset //#define umap unordered_map //#define umapi unordered_map //#define umapp unordered_map #define mmap multimap //マクロ 繰り返し #define _overloadrep(_1, _2, _3, _4, name, ...) name # define _rep(i, n) for(int i = 0,_lim=n; i < _lim ; i++) #define repi(i, m, n) for(int i = m,_lim=n; i < _lim ; i++) #define repadd(i, m, n, ad) for(int i = m,_lim=n; i < _lim ; i+= ad) #define rep(...) _overloadrep(__VA_ARGS__,repadd,repi,_rep,)(__VA_ARGS__) #define _rer(i, n) for(int i = n; i >= 0 ; i--) #define reri(i, m, n) for(int i = m,_lim=n; i >= _lim ; i--) #define rerdec(i, m, n, dec) for(int i = m,_lim=n; i >= _lim ; i-=dec) #define rer(...) _overloadrep(__VA_ARGS__,rerdec,reri,_rer,)(__VA_ARGS__) #define fora(a, b) for(auto&& a : b) //マクロ 定数 #define k3 1010 #define k4 10101 #define k5 101010 #define k6 1010101 #define k7 10101010 const int inf = (int) 1e9 + 100; const ll linf = (ll) 1e18 + 100; const double eps = 1e-9; const double PI = 3.1415926535897932384626433832795029L; ll ma = numeric_limits::min(); ll mi = numeric_limits::max(); const int y4[] = {-1, 1, 0, 0}; const int x4[] = {0, 0, -1, 1}; const int y8[] = {0, 1, 0, -1, -1, 1, 1, -1}; const int x8[] = {1, 0, -1, 0, 1, -1, 1, -1}; //マクロ省略形 関数等 #define arsz(a) (sizeof(a)/sizeof(a[0])) #define sz(a) ((int)(a).size()) #define rs resize #define mp make_pair #define pb push_back #define pf push_front #define eb emplace_back #define all(a) (a).begin(),(a).end() #define rall(a) (a).rbegin(),(a).rend() inline void sort(string &a) { sort(a.begin(), a.end()); } template inline void sort(vector &a) { sort(a.begin(), a.end()); }; template inline void sort(vector &a, int len) { sort(a.begin(), a.begin() + len); }; template inline void sort(vector &a, F f) { sort(a.begin(), a.end(), [&](T l, T r) { return f(l) < f(r); }); }; enum ___pcomparator { fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd }; inline void sort(vector

&a, ___pcomparator type) { switch (type) { case fisi: sort(all(a), [&](P l, P r) { return l.fi != r.fi ? l.fi < r.fi : l.se < r.se; }); break; case fisd: sort(all(a), [&](P l, P r) { return l.fi != r.fi ? l.fi < r.fi : l.se > r.se; }); break; case fdsi: sort(all(a), [&](P l, P r) { return l.fi != r.fi ? l.fi > r.fi : l.se < r.se; }); break; case fdsd: sort(all(a), [&](P l, P r) { return l.fi != r.fi ? l.fi > r.fi : l.se > r.se; }); break; case sifi: sort(all(a), [&](P l, P r) { return l.se != r.se ? l.se < r.se : l.fi < r.fi; }); break; case sifd: sort(all(a), [&](P l, P r) { return l.se != r.se ? l.se < r.se : l.fi > r.fi; }); break; case sdfi: sort(all(a), [&](P l, P r) { return l.se != r.se ? l.se > r.se : l.fi < r.fi; }); break; case sdfd: sort(all(a), [&](P l, P r) { return l.se != r.se ? l.se > r.se : l.fi > r.fi; }); break; } }; inline void sort(vector &a, ___pcomparator type) { switch (type) { case fisi: sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f < r.f : l.s < r.s; }); break; case fisd: sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f < r.f : l.s > r.s; }); break; case fdsi: sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f > r.f : l.s < r.s; }); break; case fdsd: sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f > r.f : l.s > r.s; }); break; case sifi: sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s < r.s : l.f < r.f; }); break; case sifd: sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s < r.s : l.f > r.f; }); break; case sdfi: sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s > r.s : l.f < r.f; }); break; case sdfd: sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s > r.s : l.f > r.f; }); break; } }; template inline void rsort(vector &a) { sort(a.begin(), a.end(), greater()); }; template inline void rsort(vector &a, int len) { sort(a.begin(), a.begin() + len, greater()); }; template inline void rsort(vector &a, F f) { sort(a.begin(), a.end(), [&](U l, U r) { return f(l) > f(r); }); }; template inline void sortp(vector &a, vector &b) { vp c; int n = sz(a); assert(n == sz(b)); rep(i, n)c.eb(a[i], b[i]); sort(c); rep(i, n) { a[i] = c[i].first; b[i] = c[i].second;; } }; //F = T //例えばreturn p.fi + p.se; template inline void sortp(vector &a, vector &b, F f) { vp c; int n = sz(a); assert(n == sz(b)); rep(i, n)c.eb(a[i], b[i]); sort(c, f); rep(i, n) { a[i] = c[i].first; b[i] = c[i].second; } }; template inline void sortp(vector &a, vector &b, char type) { vp c; int n = sz(a); assert(n == sz(b)); rep(i, n)c.eb(a[i], b[i]); sort(c, type); rep(i, n) { a[i] = c[i].first; b[i] = c[i].second; } }; template inline void rsortp(vector &a, vector &b) { vp c; int n = sz(a); assert(n == sz(b)); rep(i, n)c.eb(a[i], b[i]); rsort(c); rep(i, n) { a[i] = c[i].first; b[i] = c[i].second; } }; template inline void rsortp(vector &a, vector &b, F f) { vp c; int n = sz(a); assert(n == sz(b)); rep(i, n)c.eb(a[i], b[i]); rsort(c, f); rep(i, n) { a[i] = c[i].first; b[i] = c[i].second; } }; template inline void sortt(vector &a, vector &b, vector &c) { vt r; int n = sz(a); assert(n == sz(b)); assert(n == sz(c)); rep(i, n)r.eb(a[i], b[i], c[i]); sort(r); rep(i, n) { a[i] = r[i].f; b[i] = r[i].s; c[i] = r[i].t; } }; template inline void sortt(vector &a, vector &b, vector &c, F f) { vt r; int n = sz(a); assert(n == sz(b)); assert(n == sz(c)); rep(i, n)r.eb(a[i], b[i], c[i]); sort(r, f); rep(i, n) { a[i] = r[i].f; b[i] = r[i].s; c[i] = r[i].t; } }; template inline void rsortt(vector &a, vector &b, vector &c, F f) { vt r; int n = sz(a); assert(n == sz(b)); assert(n == sz(c)); rep(i, n)r.eb(a[i], b[i], c[i]); rsort(r, f); rep(i, n) { a[i] = r[i].f; b[i] = r[i].s; c[i] = r[i].t; } }; template inline void sort2(vector> &a) { for (int i = 0, n = a.size(); i < n; i++)sort(a[i]); } template inline void rsort2(vector> &a) { for (int i = 0, n = a.size(); i < n; i++)rsort(a[i]); } template void fill(A (&a)[N], const T &v) { rep(i, N)a[i] = v; } template void fill(A (&a)[N][O], const T &v) { rep(i, N)rep(j, O)a[i][j] = v; } template void fill(A (&a)[N][O][P], const T &v) { rep(i, N)rep(j, O)rep(k, P)a[i][j][k] = v; } template void fill(A (&a)[N][O][P][Q], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)a[i][j][k][l] = v; } template void fill(A (&a)[N][O][P][Q][R], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)a[i][j][k][l][m] = v; } template void fill(A (&a)[N][O][P][Q][R][S], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)rep(n, S)a[i][j][k][l][m][n] = v; } template void fill(V &xx, const T vall) { xx = vall; } template void fill(vector &vecc, const T vall) { for (auto &&vx: vecc) fill(vx, vall); } //@汎用便利関数 入力 template T _in() { T x; cin >> x; return (x); } #define _overloadin(_1, _2, _3, _4, name, ...) name #define in0() _in() #define in1(a) cin>>a #define in2(a, b) cin>>a>>b #define in3(a, b, c) cin>>a>>b>>c #define in4(a, b, c, d) cin>>a>>b>>c>>d #define in(...) _overloadin(__VA_ARGS__,in4,in3,in2 ,in1,in0)(__VA_ARGS__) #define _overloaddin(_1, _2, _3, _4, name, ...) name #define din1(a) int a;cin>>a #define din2(a, b) int a,b;cin>>a>>b #define din3(a, b, c) int a,b,c;cin>>a>>b>>c #define din4(a, b, c, d) int a,b,c,d;cin>>a>>b>>c>>d #define din(...) _overloadin(__VA_ARGS__,din4,din3,din2 ,din1)(__VA_ARGS__) #define _overloaddind(_1, _2, _3, _4, name, ...) name #define din1d(a) int a;cin>>a;a-- #define din2d(a, b) int a,b;cin>>a>>b;a--,b-- #define din3d(a, b, c) int a,b,c;cin>>a>>b>>c;a--,b--,c-- #define din4d(a, b, c, d) int a,b,c,d;cin>>a>>b>>c>>d;;a--,b--,c--,d-- #define dind(...) _overloaddind(__VA_ARGS__,din4d,din3d,din2d ,din1d)(__VA_ARGS__) string sin() { return _in(); } ll lin() { return _in(); } #define na(a, n) a.resize(n); rep(nai,n) cin >> a[nai]; #define nao(a, n) a.resize(n+1); rep(i,n) cin >> a[i+1]; #define nad(a, n)a.resize(n); rep(nadi,n){ cin >> a[nadi]; a[nadi]--;} #define na2(a, b, n) a.resize(n),b.resize(n);rep(i, n)cin >> a[i] >> b[i]; #define na2d(a, b, n) a.resize(n),b.resize(n);rep(i, n){cin >> a[i] >> b[i];a[i]--,b[i]--;} #define na3(a, b, c, n) a.resize(n),b.resize(n),c.resize(n); rep(i, n)cin >> a[i] >> b[i] >> c[i]; #define na3d(a, b, c, n) a.resize(n),b.resize(n),c.resize(n); rep(i, n){cin >> a[i] >> b[i] >> c[i];a[i]--,b[i]--,c[i]--;} #define nt(a, h, w) a.resize(h);rep(i,h)a[i].resize(w); rep(hi,h)rep(wi,w) cin >> a[hi][wi]; #define ntd(a, h, w) rs(a,h,w);rep(hi,h)rep(wi,w) cin >> a[hi][wi], a[hi][wi]--; #define ntp(a, h, w) fill(a,'#');rep(hi,1,h+1)rep(wi,1,w+1) cin >> a[hi][wi]; #ifdef _DEBUG template void err2(T &&head) { cerr << head; } template void err2(T &&head, U &&... tail) { cerr << head << " "; err2(tail...); } template void err(T &&head, U &&... tail) { cerr << head << " "; err2(tail...); cerr << "" << endl; } template void err(T &&head) { cerr << head << endl; } void err() { cerr << "" << endl; } template string out_m2(vector &a, ll W = inf) { stringstream ss; if (W == inf)W = min(sz(a), 12ll); if (sz(a) == 0)return ss.str(); rep(i, W) { ss << a[i] << " "; } return ss.str(); } template string out_m2(vector > &a, ll H = inf, ll W = inf, int key = -1) { H = min({H, sz(a), 12ll}); W = min({W, sz(a[0]), 12ll}); stringstream ss; ss << endl; if (key == -1)ss << " *|"; else ss << " " << key << "|"; rep(w, W)ss << std::right << std::setw(4) << w; ss << "" << endl; rep(w, W * 4 + 3)ss << "_"; ss << "" << endl; rep(h, H) { ss << std::right << std::setw(2) << h << "|"; rep(w, min(sz(a[h]), 12ll)) { if (abs(a[h][w]) == linf) ss << " e" << ""; else ss << std::right << std::setw(4) << a[h][w]; } ss << "" << endl; } return ss.str(); } /*@formatter:off*/ template string out_m2(vector > > &a, ll H = inf, ll W = inf, ll U = inf) {stringstream ss; if (H == inf)H = 5; H = min(H, sz(a)); rep(i, H) { ss << endl; ss << out_m2(a[i], W, U, i); } return ss.str();} string out_m2(int a) {stringstream ss;ss << a;return ss.str();} template string out_m2(T &a) {stringstream ss;ss << a;return ss.str();} template string out_m(vector &a, ll W = inf) {stringstream ss; if (W == inf)W = min(sz(a), 12ll); if(sz(a)==0)return ss.str(); rep(i, W) { ss << a[i] << " "; } ss << "" << endl; return ss.str();} template string out_m(vector > &a, ll H = inf, ll W = inf, int key = -1) { H = min({H, sz(a), 12ll}); W = min({W, sz(a[0]), 12ll}); stringstream ss; ss << endl; if (key == -1)ss << " *|"; else ss << " " << key << "|"; rep(w, W)ss << std::right << std::setw(4) << w; ss << "" << endl; rep(w, W * 4 + 3)ss << "_"; ss << "" << endl; rep(h, H) { ss << std::right << std::setw(2) << h << "|"; rep(w, min(sz(a[h]),12ll)) { if (abs(a[h][w]) == linf) ss << " e" << ""; else ss << std::right << std::setw(4) << a[h][w]; } ss << "" << endl; } ss << endl; return ss.str(); } /*@formatter:off*/ template string out_m(vector > > &a, ll H = inf, ll W = inf, ll U = inf) {stringstream ss; if (H == inf)H = 5; H = min(H, sz(a)); rep(i, H) { ss << endl; ss << out_m(a[i], W, U, i); } ss << endl; return ss.str();} string out_m(int a) {stringstream ss;ss << a << endl;return ss.str();} template string out_m(T &a) {stringstream ss;ss << a << endl;return ss.str();} template void outv(vector &a, ll W=inf) {cout << out_m(a,W) << endl;} template void outv(vector > &a, ll H = linf, ll W = linf,int key=-1) { cout << out_m(a,H,W,key) << endl;} template void outv(vector > > &a, ll H = linf, ll W = linf,ll U = linf) {cout << out_m(a,H,W,U)<< endl;} #else template void outv(vector &a, ll W = inf) { rep(i, min(W, sz(a))) { cout << a[i] << " "; } cout << "" << endl; } template void outv(vector > &a, ll H = linf, ll W = linf, int key = -1) { rep(i, min(H, sz(a))) { outv(a[i], W); }} template void outv(vector > > &a, ll H = linf, ll W = linf, ll U = linf) { ; } #define err(...); #endif #define debugName(VariableName) # VariableName #define deb1(x) debugName(x)<<" = "< void rev(vector &a) { reverse(all(a)); } void rev(string &a) { reverse(all(a)); } ll ceil(ll a, ll b) { if (b == 0) { debugline("ceil"); deb(a, b); ole(); return -1; } else return (a + b - 1) / b; } ll sqrt(ll a) { if (a < 0) { debugline("sqrt"); deb(a); ole(); } ll res = (ll) std::sqrt(a); while (res * res < a)res++; return res; } double log(double e, double x) { return log(x) / log(e); } ll sig(ll t) { return (1 + t) * t / 2; } ll sig(ll s, ll t) { return (s + t) * (t - s + 1) / 2; } /*@formatter:off*/ inline bool inside(int h, int w, int H, int W) { return h >= 0 && w >= 0 && h < H && w < W; } inline bool inside(int v, int l, int r) { return l <= v && v < r; } #define ins inside ll u(ll a) { return a < 0 ? 0 : a; } template vector u(const vector &a) { vector ret = a; fora(v, ret)v = u(v); return ret; } #define MIN(a) numeric_limits::min() #define MAX(a) numeric_limits::max() void yn(bool a) { if (a)cout << "yes" << endl; else cout << "no" << endl; } void Yn(bool a) { if (a)cout << "Yes" << endl; else cout << "No" << endl; } void YN(bool a) { if (a)cout << "YES" << endl; else cout << "NO" << endl; } void fyn(bool a) { if (a)cout << "yes" << endl; else cout << "no" << endl; exit(0); } void fYn(bool a) { if (a)cout << "Yes" << endl; else cout << "No" << endl; exit(0); } void fYN(bool a) { if (a)cout << "YES" << endl; else cout << "NO" << endl; exit(0); } void Possible(bool a) { if (a)cout << "Possible" << endl; else cout << "Impossible" << endl; exit(0); } void POSSIBLE(bool a) { if (a)cout << "POSSIBLE" << endl; else cout << "IMPOSSIBLE" << endl; exit(0); } template set &operator+=(set &a, U v) { a.insert(v); return a; } template vector &operator+=(vector &a, U v) { a.pb(v); return a; } template T sum(vector &v, int s = 0, int t = inf) { T ret = 0; rep(i, s, min(sz(v), t))ret += v[i]; return ret; } void mod(int &a, int m) { a = (a % m + m) % m; } template inline int mgr(int ok, int ng, F f) { #define _mgrbody int mid = (ok + ng) / 2; if (f(mid))ok = mid; else ng = mid; if (ok < ng)while (ng - ok > 1) { _mgrbody } else while (ok - ng > 1) { _mgrbody } return ok; } template inline int mgr(int ok, int ng, int second, F f) { #define _mgrbody2 int mid = (ok + ng) / 2; if (f(mid, second))ok = mid; else ng = mid; if (ok < ng) while (ng - ok > 1) { _mgrbody2 } else while (ok - ng > 1) { _mgrbody2 } return ok; } vp run_length(vi &a) { vp ret; ret.eb(a[0], 1); rep(i, 1, sz(a)) { if (ret.back().fi == a[i]) { ret.back().se++; } else { ret.eb(a[i], 1); }} return ret;} vector> run_length(string &a) { vector> ret; ret.eb(a[0], 1); rep(i, 1, sz(a)) { if (ret.back().fi == a[i]) { ret.back().se++; } else { ret.eb(a[i], 1); }} return ret;} template ostream &operator<<(ostream &os, vector &m) { for (auto &&v:m) os << v << " "; return os; } constexpr bool bget(ll m, int keta) { return (m >> keta) & 1; } int bget(ll m, int keta, int sinsuu) { m /= (ll) pow(sinsuu, keta); return m % sinsuu; } bool subset(int sub, int top) {return (sub & top) == sub;} ll bit(int n) { return (1LL << (n)); } ll bit(int n, int sinsuu) { return (ll) pow(sinsuu, n); } //aをbitに置きなおす //{0, 2} -> 101 ll biti(vi&a){ int m=0; for(auto&& v:a) m|=bit(v); return m;} //{1, 1, 0} -> 011 ll bitb(vi&a){ int m=0; rep(i,sz(a)) if(a[i])m|=bit(i); return m;} int mask(int n) { return (1ll << n) - 1; } #define bcou __builtin_popcountll // //imoは0-indexed //ruiは1-indexed template vector imo(vector &v) { vector ret = v; rep(i, sz(ret) - 1)ret[i + 1] += ret[i]; return ret;} //kと同じものの数 template vi imo(vector &a, U k) {vector ret = a;rep(i, sz(ret))ret[i] = a[i] == k;rep(i, sz(ret) - 1)ret[i + 1] += ret[i];return ret;} template vector imox(vector &v) { vector ret = v; rep(i, sz(ret) - 1)ret[i + 1] ^= ret[i]; return ret;} //漸化的に最小を持つ template vector imi(vector &v) { vector ret = v; rep(i, sz(ret) - 1)chmi(ret[i + 1], ret[i]); return ret;} template struct ruiC { const vector rui; ruiC(vector &ru) : rui(ru) {} T operator()(ll l, ll r) { if (l > r) { cerr<<"ruic ";deb(l, r);assert(0); } return rui[r] - rui[l]; } T operator[](ll i) { return rui[i]; } T back() { return rui.back(); } ll size() { return rui.size(); }}; template struct rruic { const T *rrui; rruic(T *ru) : rrui(ru) {} T operator()(ll l, ll r) { assert(l >= r); return rrui[r] - rrui[l]; } T operator[](ll i) { return rrui[i]; }}; templateostream &operator<<(ostream &os, ruiC a) {fora(v,a.rui)os< vector ruiv(vector &a) { vector ret(a.size() + 1); rep(i, a.size())ret[i + 1] = ret[i] + a[i]; return ret;} template ruiC ruic(vector &a) { vector ret = ruiv(a); return ruiC(ret);} vector ruiv(string &a) { if (sz(a) == 0)return vi(1); ll dec = ('0' <= a[0] && a[0] <= '9') ? '0' : 0; vector ret(a.size() + 1); rep(i, a.size())ret[i + 1] = ret[i] + a[i] - dec; return ret;} ruiC ruic(string &a) { vector ret = ruiv(a); return ruiC(ret);} //kと同じものの数 template vi ruiv(T &a, U k) { vi ret(a.size() + 1); rep(i, a.size())ret[i + 1] = ret[i] + (a[i] == k); return ret;} template ruiC ruic(T &a, U k) { vi ret = ruiv(a, k); return ruiC(ret);} //xor template struct ruixC { const vector rui; ruixC(vector &ru) : rui(ru) {} T operator()(ll l, ll r) { if (l > r) { cerr << "ruiXc "; deb(l, r); assert(0); } return rui[r] ^ rui[l]; } T operator[](ll i) { return rui[i]; } T back() { return rui.back(); } ll size() { return rui.size(); }}; template vector ruix(vector &a) { vector ret(a.size() + 1); rep(i, a.size())ret[i + 1] = ret[i] ^ a[i]; return ret;} template ruixC ruixc(vector &a) {vi ret = ruix(a);return ruixC(ret);} template vector ruim(vector &a) { vector res(a.size() + 1, 1); rep(i, a.size())res[i + 1] = res[i] * a[i]; return res;} //漸化的に最小を1indexで持つ template vector ruimi(vector &a) { ll n = sz(a); vector ret(n + 1); rep(i, 1, n) { ret[i] = a[i - 1]; chmi(ret[i + 1], ret[i]); } return ret;} //template T *rrui(vector &a) { //右から左にかけての半開区間 (-1 n-1] template rruic rrui(vector &a) { ll len = a.size(); T *body = (T *) malloc((len + 1) * sizeof(T)); T *res = body + 1; rer(i, len - 1)res[i - 1] = res[i] + a[i]; return rruic(res);} //掛け算 template T *rruim(vector &a) { ll len = a.size(); T *body = (T *) malloc((len + 1) * sizeof(T)); T *res = body + 1; res[len - 1] = 1; rer(i, len - 1)res[i - 1] = res[i] * a[i]; return res;} // template inline bool chma(T &a, const U &b) { if (a < b) { a = b; return true; } return false; } template inline bool chma(const U &b) { return chma(ma, b); } template inline bool chmi(T &a, const U &b) { if (b < a) { a = b; return true; } return false; } template inline bool chmi(const U &b) { return chmi(mi, b); } #define unique(v) v.erase( unique(v.begin(), v.end()), v.end() ); int max(vi &a) { int res = a[0]; fora(v, a) { res = max(res, v); } return res; } int min(vi &a) { int res = a[0]; fora(v, a) { res = min(res, v); } return res; } template class fixed_point : T {public: explicit constexpr fixed_point(T &&t) noexcept: T(std::forward(t)) {} template constexpr decltype(auto) operator()(Args &&... args) const { return T::operator()(*this, std::forward(args)...); }};template static inline constexpr decltype(auto) fix(T &&t) noexcept { return fixed_point{std::forward(t)}; } #define lam(right) [&](auto& p){return p right;} template void resize(vector &vec, const H head) { vec.resize(head); }template void resize(vector &vec, const H &head, const T ... tail) {vec.resize(head);for (auto &v: vec)resize(v, tail...);} template bool all_of2(T &v, F f) { return f(v); } template bool all_of2(vector &v, F f) { rep(i, sz(v)) { if (!all_of2(v[i], f))return false; } return true;} template bool any_of2(T &v, F f) { return f(v); } template bool any_of2(vector &v, F f) { rep(i, sz(v)) { if (any_of2(v[i], f))return true; } return false;} template bool none_of2(T &v, F f) { return f(v); } template bool none_of2(vector &v, F f) { rep(i, sz(v)) { if (none_of2(v[i], f))return false; } return true;} template bool find_if2(T &v, F f) { return f(v); } template ll find_if2(vector &v, F f) { rep(i, sz(v)) { if (find_if2(v[i], f))return i; } return sz(v);} template bool rfind_if2(T &v, F f) { return f(v); } template ll rfind_if2(vector &v, F f) { rer(i, sz(v) - 1) { if (rfind_if2(v[i], f))return i; } return -1;} template bool contains(string &s, const T &v) { return s.find(v) != string::npos; } template bool contains(vector &v, const T &val) { return std::find(v.begin(), v.end(), val) != v.end(); } template bool contains_if2(vector &v, F f) { return find_if(v.begin(), v.end(), f) != v.end(); } template ll count_if2(T &v, F f) { return f(v); } template ll count_if2(vector &vec, F f) { ll ret = 0; fora(v, vec)ret += count_if2(v, f); return ret;} template void for_each2(T &v, F f) { f(v); } template void for_each2(vector &vec, F f) { fora(v, vec)for_each2(v, f); } template ll count_od(vector &a) {return count_if2(a,[](ll v){return v&1 ;});} template ll count_ev(vector &a) {return count_if2(a,[](ll v){return !(v&1) ;});} //削除された要素の数を返す template int erase_if2(vector &v, F f) {vector nv;int cou = 0;rep(i, sz(v)) {if (f(v[i])) {cou++;} else {nv.push_back(v[i]);}}v = nv;return cou;} template int erase_if2(vector> &v, F f) {int cou = 0;rep(i, sz(v)) {cou += erase_if2(v[i], f);}return cou;} #define all_of(a,right) all_of2(a,lam(right)) #define all_of_f(a,f) all_of2(a,f) #define any_of(a,right) any_of2(a,lam(right)) #define any_of_f(a,f) any_of2(a,f) #define none_of(a,right) none_of2(a,lam(right)) #define none_of_f(a,f) none_of2(a,f) #define find_if(a,right) find_if2(a,lam(right)) #define find_if_f(a,f) find_if2(a,f) #define rfind_if(a,right) rfind_if2(a,lam(right)) #define rfind_if_f(a,f) rfind_if2(a,f) #define contains_if(a,right) contains_if2(a,lam(right)) #define contains_if_f(a,f) contains_if2(a,f) #define count_if(a, right) count_if2(a,lam(right)) #define count_if_f(a, f) count_if2(a,f) #define for_each(a, right) do{fora(v,a){v right;}}while(0) #define for_each_f(a, f) do{fora(v,a){f(v);}}while(0) #define erase_if(a,right) erase_if2(a,lam(right)) #define erase_if_f(a,f) erase_if2(a,f) #define dna(a, n) vi a(n); rep(dnai,n) cin >> a[dnai]; #define binarySearch(a, v) (binary_search(all(a),v)) #define lowerIndex(a, v) (lower_bound(all(a),v)-a.begin()) #define upperIndex(a, v) (upper_bound(all(a),v)-a.begin()) #define rlowerIndex(a, v) (upper_bound(all(a),v)-a.begin()-1) #define rupperIndex(a, v) (lower_bound(all(a),v)-a.begin()-1) template T lowerBound(vector& a, U v, W banpei){ auto it = lower_bound(a.begin(),a.end(),v); if(it==a.end())return banpei; else return *it;} template T upperBound(vector& a, U v, W banpei){ auto it = upper_bound(a.begin(),a.end(),v); if(it==a.end())return banpei; else return *it;} template T rlowerBound(vector &a, U v, W banpei) { auto it = upper_bound(a.begin(), a.end(), v); if (it == a.begin())return banpei; else{ return *(--it); }} template T rupperBound(vector &a, U v, W banpei) { auto it = lower_bound(a.begin(), a.end(), v); if (it == a.begin())return banpei; else{ return *(--it); }} /*@formatter:on*/ template T min(vector> &a) { T res = MAX(T); rep(i, a.size())chmi(res, *min_element(all(a[i]))); return res; } template T max(vector> &a) { T res = MIN(T); rep(i, a.size())chma(res, *max_element(all(a[i]))); return res; } template T min(vector >> &a) { T res = MAX(T); rep(i, a.size())chmi(res, min(a[i])); return res; } template T max(vector >> &a) { T res = MIN(T); rep(i, a.size())chma(res, max(a[i])); return res; } template void out2(T &&head) { cout << head; } template void out2(T &&head, U &&... tail) { cout << head << " "; out2(tail...); } template void out(T &&head, U &&... tail) { cout << head << " "; out2(tail...); cout << "" << endl; } template void out(T &&head) { cout << head << endl; } void out() { cout << "" << endl; } template void fin(T a) { out(a); exit(0); } template void insert(vector &v, unsigned ll i, U t) { v.insert(v.begin() + i, t); } int N, K, H, W; vi A, B, C; //@formatter:off template T minv(T a, T m); template T minv(T a); template class Modular { public: using Type = typename decay::type; constexpr Modular() : value() {} template Modular(const U &x) { value = normalize(x); } template static Type normalize(const U &x) { Type v; if (-mod() <= x && x < mod()) v = static_cast(x); else v = static_cast(x % mod()); if (v < 0) v += mod(); return v; } const Type &operator()() const { return value; } templateexplicit operator U() const { return static_cast(value); } constexpr static Type mod() { return T::value; } Modular &operator+=(const Modular &other) { if ((value += other.value) >= mod()) value -= mod(); return *this; } Modular &operator-=(const Modular &other) { if ((value -= other.value) < 0) value += mod(); return *this; } template Modular &operator+=(const U &other) { return *this += Modular(other); } template Modular &operator-=(const U &other) { return *this -= Modular(other); } Modular &operator++() { return *this += 1; } Modular &operator--() { return *this -= 1; } Modular operator++(signed) { Modular result(*this); *this += 1; return result; } Modular operator--(signed) { Modular result(*this); *this -= 1; return result; } Modular operator-() const { return Modular(-value); } templatetypename enable_if::Type, signed>::value, Modular>::type &operator*=(const Modular &rhs) { #ifdef _WIN32 uint64_t x = static_cast(value) * static_cast(rhs.value);uint32_t xh = static_cast(x >> 32), xl = static_cast(x), d, m;asm("divl %4; \n\t": "=a" (d), "=d" (m): "d" (xh), "a" (xl), "r" (mod()));value = m; #else value = normalize(static_cast(value) * static_cast(rhs.value)); #endif return *this; } template typename enable_if::Type, int64_t>::value, Modular>::type &operator*=(const Modular &rhs) { int64_t q = static_cast(static_cast(value) * rhs.value / mod()); value = normalize(value * rhs.value - q * mod()); return *this; } template typename enable_if::Type>::value, Modular>::type &operator*=(const Modular &rhs) { value = normalize(value * rhs.value); return *this; } Modular &operator/=(const Modular &other) { return *this *= Modular(minv(other.value)); } template friend bool operator==(const Modular &lhs, const Modular &rhs); template friend bool operator<(const Modular &lhs, const Modular &rhs); template friend std::istream &operator>>(std::istream &stream, Modular &number); operator int() { return value; }private: Type value; }; template bool operator==(const Modular &lhs, const Modular &rhs) { return lhs.value == rhs.value; }template bool operator==(const Modular &lhs, U rhs) { return lhs == Modular(rhs); }template bool operator==(U lhs, const Modular &rhs) { return Modular(lhs) == rhs; }template bool operator!=(const Modular &lhs, const Modular &rhs) { return !(lhs == rhs); }template bool operator!=(const Modular &lhs, U rhs) { return !(lhs == rhs); }template bool operator!=(U lhs, const Modular &rhs) { return !(lhs == rhs); }template bool operator<(const Modular &lhs, const Modular &rhs) { return lhs.value < rhs.value; }template Modular operator+(const Modular &lhs, const Modular &rhs) { return Modular(lhs) += rhs; }template Modular operator+(const Modular &lhs, U rhs) { return Modular(lhs) += rhs; }template Modular operator+(U lhs, const Modular &rhs) { return Modular(lhs) += rhs; }template Modular operator-(const Modular &lhs, const Modular &rhs) { return Modular(lhs) -= rhs; }template Modular operator-(const Modular &lhs, U rhs) { return Modular(lhs) -= rhs; }template Modular operator-(U lhs, const Modular &rhs) { return Modular(lhs) -= rhs; }template Modular operator*(const Modular &lhs, const Modular &rhs) { return Modular(lhs) *= rhs; }template Modular operator*(const Modular &lhs, U rhs) { return Modular(lhs) *= rhs; }template Modular operator*(U lhs, const Modular &rhs) { return Modular(lhs) *= rhs; }template Modular operator/(const Modular &lhs, const Modular &rhs) { return Modular(lhs) /= rhs; }template Modular operator/(const Modular &lhs, U rhs) { return Modular(lhs) /= rhs; }template Modular operator/(U lhs, const Modular &rhs) { return Modular(lhs) /= rhs; } constexpr signed MOD = // 998244353; 1e9 + 7;//MOD using mint = Modular::type, MOD>>; constexpr int mint_len = 1400001; vi fac, finv, inv; vi p2; mint com(int n, int r) { if (r < 0 || r > n) return 0; return mint(finv[r] * fac[n] % MOD * finv[n - r]);} mint pom(int n, int r) {/* if (!sz(fac)) com(0, -1);*/ if (r < 0 || r > n) return 0; return mint(fac[n] * finv[n - 1]);} mint npr(int n, int r) {/* if (!sz(fac)) com(0, -1);*/ if (r < 0 || r > n) return 0; return mint(fac[n] * finv[n - r]);} int nprin(int n, int r) {/* if (!sz(fac)) com(0, -1);*/ if (r < 0 || r > n) return 0; return fac[n] * finv[n - r] % MOD;} int icom(int n, int r) { const int NUM_ = 1400001; static ll fac[NUM_ + 1], finv[NUM_ + 1], inv[NUM_ + 1]; if (fac[0] == 0) { inv[1] = fac[0] = finv[0] = 1; for (int i = 2; i <= NUM_; ++i) inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD; for (int i = 1; i <= NUM_; ++i) fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD; } if (r < 0 || r > n) return 0; return ((finv[r] * fac[n] % MOD) * finv[n - r]) % MOD;} #define ncr com #define ncri icom //n個の場所にr個の物を置く mint nhr(int n, int r) { return com(n + r - 1, r); } mint hom(int n, int r) { return com(n + r - 1, r); } int nhri(int n, int r) { return icom(n + r - 1, r); } template T minv(T a, T m) { T u = 0, v = 1; while (a != 0) { T t = m / a; m -= t * a; swap(a, m); u -= t * v; swap(u, v); } assert(m == 1); return u;} template T minv(T a) { if (a < mint_len)return inv[a]; T u = 0, v = 1; T m = MOD; while (a != 0) { T t = m / a; m -= t * a; swap(a, m); u -= t * v; swap(u, v); } assert(m == 1); return u;} template Modular mpow(const Modular &a, const U &b) { assert(b >= 0); int x = a(), res = 1; U p = b; while (p > 0) { if (p & 1) (res *= x) %= MOD; (x *= x) %= MOD; p >>= 1; } return res;} template mint mpow(const T a, const U b, const V m = MOD) { assert(b >= 0); int x = a, res = 1; U p = b; while (p > 0) { if (p & 1) (res *= x) %= m; (x *= x) %= m; p >>= 1; } return res;} //-k乗出来る template mint mpow(const T a, const U b) {/* assert(b >= 0);*/ if(b<0){ return minv(mpow(a,-b)); } int x = a, res = 1; U p = b; while (p > 0) { if (p & 1) (res *= x) %= MOD; (x *= x) %= MOD; p >>= 1; } return res;} template int mpowi(const T &a, const U &b, const V &m = MOD) { assert(b >= 0); int x = a, res = 1; U p = b; while (p > 0) { if (p & 1) (res *= x) %= m; (x *= x) %= m; p >>= 1; } return res;} template string to_string(const Modular &number) { return to_string(number());} #ifdef _DEBUG void yuri(const mint &a) { stringstream st; rep(i, 300) { rep(j, 300) { if ((mint) i / j == a) { st << i << " / " << j; i=2000;break;}}} string val = st.str(); if(val!=""){ deb(val); return; } rep(i, 1000) { rep(j, 1000) { if ((mint) i / j == a) { st << i << " / " << j;i=2000; break;}}} val = st.str(); deb(val);} #else #define yuri(...) ; #endif template std::ostream &operator<<(std::ostream &stream, const Modular &number) {stream << number(); #ifdef _DEBUG // stream << " -> " << yuri(number); #endif return stream; } //@formatter:off template std::istream &operator>>(std::istream &stream, Modular &number) { typename common_type::Type, int64_t>::type x; stream >> x; number.value = Modular::normalize(x); return stream;} using PM = pair; using vm = vector; using mapm = map; //using umapm = umap; ll max(int a, ll b){ if(a(__VA_ARGS__) string out_m2(mint a) { stringstream st; rep(i, 300) { rep(j, 300) { if ((mint) i / j == a) { st << i << " / " << j; i = 2000; break; } } } string val = st.str(); // if (val != "") { return val; } // rep(i, 1000) { // rep(j, 1000) { // if ((mint) i / j == a) { // st << i << " / " << j; // i = 2000; // break; // } // } // } val = st.str(); return val; } struct setmod{ setmod() { p2.resize(mint_len);p2[0] = 1; for (int i = 1; i < mint_len; ++i) p2[i] = p2[i - 1] * 2 % MOD; fac.resize(mint_len); finv.resize(mint_len); inv.resize(mint_len); inv[1] = fac[0] = finv[0] = 1; for (int i = 2; i < mint_len; ++i) inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD; for (int i = 1; i < mint_len; ++i) fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD; } }setmodv; //@formatter:on //nhr n個の場所にr個の物を分ける /*@formatter:off*/ mint m1 = (mint)1; mint half = (mint)1 / 2; template< typename T > struct FormalPowerSeries : vector< T > { using vector< T >::vector; using P = FormalPowerSeries; using MULT = function< P(P, P) >; static MULT &get_mult() { static MULT mult = nullptr; return mult; } static void set_fft(MULT f) { get_mult() = f; } FormalPowerSeries(const vector< T > &v) : FormalPowerSeries(v.begin(), v.end()) {} P operator+(const P &r) const { return P(*this) += r; } P operator-(const P &r) const { return P(*this) -= r; } P operator*(const P &r) const { return P(*this) *= r; } P operator*(const T &v) const { return P(*this) *= v; } P operator/(const P &r) const { return P(*this) /= r; } P &operator+=(const P &r) { if(r.size() > this->size()) this->resize(r.size()); for(signed i = 0; i < r.size(); i++) (*this)[i] += r[i]; return *this; } P &operator-=(const P &r) { if(r.size() > this->size()) this->resize(r.size()); for(signed i = 0; i < r.size(); i++) (*this)[i] -= r[i]; return *this; } P &operator*=(const T &v) { const signed n = (signed) this->size(); for(signed k = 0; k < n; k++) (*this)[k] *= v; return *this; } P &operator*=(const P &r) { if(this->empty() || r.empty()) { this->clear(); return *this; } assert(get_mult() != nullptr); auto ret = get_mult()(*this, r); this->resize(ret.size()); for(signed k = 0; k < ret.size(); k++) (*this)[k] = ret[k]; return *this; } P operator-() const { P ret(this->size()); for(signed i = 0; i < this->size(); i++) ret[i] = -(*this)[i]; return ret; } P &operator/=(const P &r) { return *this *= r.inverse(); } P pre(signed sz) const { return P(begin(*this), begin(*this) + min((signed) this->size(), sz)); } P rev() const { P ret(*this); reverse(begin(ret), end(ret)); return ret; } P diff() const { const signed n = (signed) this->size(); P ret(max(0, n - 1)); for(signed i = 1; i < n; i++) ret[i] = (*this)[i] * T(i); return ret; } P integral() const { const signed n = (signed) this->size(); P ret(n + 1); ret[0] = T(0); for(signed i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1); } // F(0) must not be 0 P inv(signed deg = -1) const { assert(((*this)[0]) != T(0)); const signed n = (signed) this->size(); if(deg == -1) deg = n; P ret({T(1) / (*this)[0]}); for(signed i = 1; i < deg; i <<= 1) { ret = (ret * T(2) - ret * ret * pre(i << 1)).pre(i << 1); } return ret.pre(deg); } // F(0) must be 1 P log(signed deg = -1) const { assert((*this)[0] == 1); const signed n = (signed) this->size(); if(deg == -1) deg = n; return (this->diff() * this->inv(deg)).integral().pre(deg); } // F(0) must be 1 P sqrt(signed deg = -1) const { assert((*this)[0] == T(1)); const signed n = (signed) this->size(); if(deg == -1) deg = n; P ret({T(1)}); T inv2 = T(1) / T(2); for(signed i = 1; i < deg; i <<= 1) { ret = (ret + pre(i << 1) * ret.inv(i << 1)).pre(i << 1) * 2; } return ret.pre(deg); } // F(0) must be 0 P exp(signed deg) const { assert((*this)[0] == T(0)); const signed n = (signed) this->size(); P ret({T(1)}), g({T(1)}); for(signed i = 1; i < deg; i <<= 1) { ret = (ret * (pre(1 << i) + g) - ret.log(1 << i)).pre(1 << i); } return ret.pre(deg); } }; namespace FastFourierTransform { using real = double; struct C { real x, y; C() : x(0), y(0) {} C(real x, real y) : x(x), y(y) {} inline C operator+(const C &c) const { return C(x + c.x, y + c.y); } inline C operator-(const C &c) const { return C(x - c.x, y - c.y); } inline C operator*(const C &c) const { return C(x * c.x - y * c.y, x * c.y + y * c.x); } inline C conj() const { return C(x, -y); } }; const real PI = acosl(-1); signed base = 1; vector< C > rts = {{0, 0}, {1, 0}}; vector< signed > rev = {0, 1}; void ensure_base(signed nbase) { if(nbase <= base) return; rev.resize(1 << nbase); rts.resize(1 << nbase); for(signed i = 0; i < (1 << nbase); i++) { rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1)); } while(base < nbase) { real angle = PI * 2.0 / (1 << (base + 1)); for(signed i = 1 << (base - 1); i < (1 << base); i++) { rts[i << 1] = rts[i]; real angle_i = angle * (2 * i + 1 - (1 << base)); rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i)); } ++base; } } void fft(vector< C > &a, signed n) { assert((n & (n - 1)) == 0); signed zeros = __builtin_ctz(n); ensure_base(zeros); signed shift = base - zeros; for(signed i = 0; i < n; i++) { if(i < (rev[i] >> shift)) { swap(a[i], a[rev[i] >> shift]); } } for(signed k = 1; k < n; k <<= 1) { for(signed i = 0; i < n; i += 2 * k) { for(signed j = 0; j < k; j++) { C z = a[i + j + k] * rts[j + k]; a[i + j + k] = a[i + j] - z; a[i + j] = a[i + j] + z; } } } } vector< int64_t > multiply(const vector< signed > &a, const vector< signed > &b) { signed need = (signed) a.size() + (signed) b.size() - 1; signed nbase = 1; while((1 << nbase) < need) nbase++; ensure_base(nbase); signed sz = 1 << nbase; vector< C > fa(sz); for(signed i = 0; i < sz; i++) { signed x = (i < (signed) a.size() ? a[i] : 0); signed y = (i < (signed) b.size() ? b[i] : 0); fa[i] = C(x, y); } fft(fa, sz); C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0); for(signed i = 0; i <= (sz >> 1); i++) { signed j = (sz - i) & (sz - 1); C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r; fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r; fa[i] = z; } for(signed i = 0; i < (sz >> 1); i++) { C A0 = (fa[i] + fa[i + (sz >> 1)]) * t; C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * rts[(sz >> 1) + i]; fa[i] = A0 + A1 * s; } fft(fa, sz >> 1); vector< int64_t > ret(need); for(signed i = 0; i < need; i++) { ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x); } return ret; } }; template< typename T > struct ArbitraryModConvolution { using real = FastFourierTransform::real; using C = FastFourierTransform::C; ArbitraryModConvolution() = default; vector< T > multiply(const vector< T > &a, const vector< T > &b, signed need = -1) { if(need == -1) need = a.size() + b.size() - 1; signed nbase = 0; while((1 << nbase) < need) nbase++; FastFourierTransform::ensure_base(nbase); signed sz = 1 << nbase; vector< C > fa(sz); for(signed i = 0; i < a.size(); i++) { fa[i] = C((signed)a[i] & ((1 << 15) - 1), (signed)a[i] >> 15); } fft(fa, sz); vector< C > fb(sz); if(a == b) { fb = fa; } else { for(signed i = 0; i < b.size(); i++) { fb[i] = C((signed)b[i] & ((1 << 15) - 1), (signed)b[i] >> 15); } fft(fb, sz); } real ratio = 0.25 / sz; C r2(0, -1), r3(ratio, 0), r4(0, -ratio), r5(0, 1); for(signed i = 0; i <= (sz >> 1); i++) { signed j = (sz - i) & (sz - 1); C a1 = (fa[i] + fa[j].conj()); C a2 = (fa[i] - fa[j].conj()) * r2; C b1 = (fb[i] + fb[j].conj()) * r3; C b2 = (fb[i] - fb[j].conj()) * r4; if(i != j) { C c1 = (fa[j] + fa[i].conj()); C c2 = (fa[j] - fa[i].conj()) * r2; C d1 = (fb[j] + fb[i].conj()) * r3; C d2 = (fb[j] - fb[i].conj()) * r4; fa[i] = c1 * d1 + c2 * d2 * r5; fb[i] = c1 * d2 + c2 * d1; } fa[j] = a1 * b1 + a2 * b2 * r5; fb[j] = a1 * b2 + a2 * b1; } fft(fa, sz); fft(fb, sz); vector< T > ret(need); for(signed i = 0; i < need; i++) { int64_t aa = llround(fa[i].x); int64_t bb = llround(fb[i].x); int64_t cc = llround(fa[i].y); aa = T(aa), bb = T(bb), cc = T(cc); ret[i] = aa + (bb << 15) + (cc << 30); } return ret; } }; signed main() { ArbitraryModConvolution< mint> fft; using FPS = FormalPowerSeries< mint >; auto mult = [&](const FPS::P &a, const FPS::P &b) { return fft.multiply(a, b); }; FPS::set_fft(mult); int N, K; cin >> K >> N; FPS X(K + 1); X[0] = 1; for(int i = 0; i < N; i++) { int x; cin >> x; if(x <= K) X[x] = -1; } cout << X.inv(K + 1)[K] << endl; return 0; }