#pragma region Macros #pragma comment(linker, "/stack:400000000") #if defined(noimi) && defined(_GLIBCXX_DEBUG) && defined(_GLIBCXX_DEBUG_PEDANTIC) #include #pragma GCC optimize("O3") #else #pragma GCC optimize("O2") // #pragma GCC optimize("unroll-loops") // #pragma GCC target("popcnt") // #pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,fma,abm,mmx,avx,avx2,tune=native") // #pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,fma,abm,mmx,avx,avx2") // #pragma GCC target("avx2") // #include // #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #endif #include #include #include #ifdef noimi #define oj_local(a, b) b #else #define oj_local(a, b) a #endif #define LOCAL if(oj_local(0, 1)) #define OJ if(oj_local(1, 0)) using namespace std; using ll = long long; using ull = unsigned long long int; using i128 = __int128_t; using pii = pair; using pll = pair; using ld = long double; template using vc = vector; template using vvc = vector>; template using vvvc = vector>; using vi = vc; using vl = vc; using vpi = vc; using vpl = vc; template using pq = priority_queue; template using pqg = priority_queue, greater>; template int si(const T &x) { return x.size(); } template inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } vi iota(int n) { vi a(n); return iota(a.begin(), a.end(), 0), a; } template vi iota(const vector &a, bool greater = false) { vi res(a.size()); iota(res.begin(), res.end(), 0); sort(res.begin(), res.end(), [&](int i, int j) { if(greater) return a[i] > a[j]; return a[i] < a[j]; }); return res; } // macros #define overload5(a, b, c, d, e, name, ...) name #define overload4(a, b, c, d, name, ...) name #define endl '\n' #define REP0(n) for(ll jidlsjf = 0; jidlsjf < n; ++jidlsjf) #define REP1(i, n) for(ll i = 0; i < (n); ++i) #define REP2(i, a, b) for(ll i = (a); i < (b); ++i) #define REP3(i, a, b, c) for(ll i = (a); i < (b); i += (c)) #define rep(...) overload4(__VA_ARGS__, REP3, REP2, REP1, REP0)(__VA_ARGS__) #define per0(n) for(int jidlsjf = 0; jidlsjf < (n); ++jidlsjf) #define per1(i, n) for(ll i = (n)-1; i >= 0; --i) #define per2(i, a, b) for(ll i = (a)-1; i >= b; --i) #define per3(i, a, b, c) for(ll i = (a)-1; i >= (b); i -= (c)) #define per(...) overload4(__VA_ARGS__, per3, per2, per1, per0)(__VA_ARGS__) #define fore0(a) rep(a.size()) #define fore1(i, a) for(auto &&i : a) #define fore2(a, b, v) for(auto &&[a, b] : v) #define fore3(a, b, c, v) for(auto &&[a, b, c] : v) #define fore4(a, b, c, d, v) for(auto &&[a, b, c, d] : v) #define fore(...) overload5(__VA_ARGS__, fore4, fore3, fore2, fore1, fore0)(__VA_ARGS__) #define fors(s, b) for(int s = (b); s >= 0; (s ? (s - 1) & b : -1)) #define fi first #define se second #define pb push_back #define ppb pop_back #define ppf pop_front #define eb emplace_back #define drop(s) cout << #s << endl, exit(0) #define si(c) (int)(c).size() #define lb(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define ub(c, x) distance((c).begin(), upper_bound(all(c), (x))) #define rng(v, l, r) v.begin() + (l), v.begin() + (r) #define all(c) begin(c), end(c) #define rall(c) rbegin(c), rend(c) #define SORT(v) sort(all(v)) #define REV(v) reverse(all(v)) #define UNIQUE(x) SORT(x), x.erase(unique(all(x)), x.end()) template T SUM(const S &v) { return accumulate(all(v), T(0)); } #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define overload2(_1, _2, name, ...) name #define vec(type, name, ...) vector name(__VA_ARGS__) #define vv(type, name, h, ...) vector> name(h, vector(__VA_ARGS__)) #define vvv(type, name, h, w, ...) vector>> name(h, vector>(w, vector(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector>>> name(a, vector>>(b, vector>(c, vector(__VA_ARGS__)))) constexpr pii dx4[4] = {pii{1, 0}, pii{0, 1}, pii{-1, 0}, pii{0, -1}}; constexpr pii dx8[8] = {{1, 0}, {1, 1}, {0, 1}, {-1, 1}, {-1, 0}, {-1, -1}, {0, -1}, {1, -1}}; namespace yesno_impl { const string YESNO[2] = {"NO", "YES"}; const string YesNo[2] = {"No", "Yes"}; const string yesno[2] = {"no", "yes"}; const string firstsecond[2] = {"second", "first"}; const string FirstSecond[2] = {"Second", "First"}; const string possiblestr[2] = {"impossible", "possible"}; const string Possiblestr[2] = {"Impossible", "Possible"}; void YES(bool t = 1) { cout << YESNO[t] << endl; } void NO(bool t = 1) { YES(!t); } void Yes(bool t = 1) { cout << YesNo[t] << endl; } void No(bool t = 1) { Yes(!t); } void yes(bool t = 1) { cout << yesno[t] << endl; } void no(bool t = 1) { yes(!t); } void first(bool t = 1) { cout << firstsecond[t] << endl; } void First(bool t = 1) { cout << FirstSecond[t] << endl; } void possible(bool t = 1) { cout << possiblestr[t] << endl; } void Possible(bool t = 1) { cout << Possiblestr[t] << endl; } }; // namespace yesno_impl using namespace yesno_impl; #define INT(...) \ int __VA_ARGS__; \ IN(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ IN(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ IN(__VA_ARGS__) #define CHR(...) \ char __VA_ARGS__; \ IN(__VA_ARGS__) #define DBL(...) \ double __VA_ARGS__; \ IN(__VA_ARGS__) #define VEC(type, name, size) \ vector name(size); \ IN(name) #define VEC2(type, name1, name2, size) \ vector name1(size), name2(size); \ for(int i = 0; i < size; i++) IN(name1[i], name2[i]) #define VEC3(type, name1, name2, name3, size) \ vector name1(size), name2(size), name3(size); \ for(int i = 0; i < size; i++) IN(name1[i], name2[i], name3[i]) #define VEC4(type, name1, name2, name3, name4, size) \ vector name1(size), name2(size), name3(size), name4(size); \ for(int i = 0; i < size; i++) IN(name1[i], name2[i], name3[i], name4[i]); #define VV(type, name, h, w) \ vector> name(h, vector(w)); \ IN(name) int scan() { return getchar(); } void scan(int &a) { cin >> a; } void scan(long long &a) { cin >> a; } void scan(char &a) { cin >> a; } void scan(double &a) { cin >> a; } void scan(string &a) { cin >> a; } template void scan(pair &p) { scan(p.first), scan(p.second); } template void scan(vector &); template void scan(vector &a) { for(auto &i : a) scan(i); } template void scan(T &a) { cin >> a; } void IN() {} template void IN(Head &head, Tail &...tail) { scan(head); IN(tail...); } template T ceil(T x, S y) { assert(y); return (y < 0 ? ceil(-x, -y) : (x > 0 ? (x + y - 1) / y : x / y)); } template T floor(T x, S y) { assert(y); return (y < 0 ? floor(-x, -y) : (x > 0 ? x / y : x / y - (x % y == 0 ? 0 : 1))); } template T POW(T x, int n) { T res = 1; for(; n; n >>= 1, x *= x) if(n & 1) res *= x; return res; } template T POW(T x, S n, const ll &mod) { T res = 1; x %= mod; for(; n; n >>= 1, x = x * x % mod) if(n & 1) res = res * x % mod; return res; } vector factor(ll x) { vector ans; for(ll i = 2; i * i <= x; i++) if(x % i == 0) { ans.push_back({i, 1}); while((x /= i) % i == 0) ans.back().second++; } if(x != 1) ans.push_back({x, 1}); return ans; } template vector divisor(T x) { vector ans; for(T i = 1; i * i <= x; i++) if(x % i == 0) { ans.pb(i); if(i * i != x) ans.pb(x / i); } return ans; } template void zip(vector &x) { vector y = x; UNIQUE(y); for(int i = 0; i < x.size(); ++i) { x[i] = lb(y, x[i]); } } template void fold_in(vector &v) {} template void fold_in(vector &v, Head &&a, Tail &&...tail) { for(auto e : a) v.emplace_back(e); fold_in(v, tail...); } template void renumber(vector &v) {} template void renumber(vector &v, Head &&a, Tail &&...tail) { for(auto &&e : a) e = lb(v, e); renumber(v, tail...); } template vector zip(vector &head, Args &&...args) { vector v; fold_in(v, head, args...); sort(all(v)), v.erase(unique(all(v)), v.end()); renumber(v, head, args...); return v; } template void rearrange(vector &v, const vector &id) { vector w(v.size()); rep(i, si(id)) w[i] = v[id[i]]; v.swap(w); } template vector RUI(const vector &v) { vector res(v.size() + 1); for(int i = 0; i < v.size(); i++) res[i + 1] = res[i] + v[i]; return res; } template void zeta_subset(vector &f) { int n = f.size(); for(int i = 1; i < n; i <<= 1) rep(b, n) if(!(i & b)) f[b] += f[b | i]; } template void zeta_superset(vector &f) { int n = f.size(); for(int i = 1; i < n; i <<= 1) rep(b, n) if(!(i & b)) f[b | i] += f[b]; } template void mobius_subset(vector &f) { int n = f.size(); for(int i = 1; i < n; i <<= 1) rep(b, n) if(!(i & b)) f[b] -= f[b | i]; } template void mobius_superset(vector &f) { int n = f.size(); for(int i = 1; i < n; i <<= 1) rep(b, n) if(!(i & b)) f[b | i] -= f[b]; } // 反時計周りに 90 度回転 template void rot(vector> &v) { if(empty(v)) return; int n = v.size(), m = v[0].size(); vector res(m, vector(n)); rep(i, n) rep(j, m) res[m - 1 - j][i] = v[i][j]; v.swap(res); } // x in [l, r) template bool inc(const T &x, const S &l, const S &r) { return l <= x and x < r; } // 便利関数 constexpr ll ten(int n) { return n == 0 ? 1 : ten(n - 1) * 10; } constexpr ll tri(ll n) { return n * (n + 1) / 2; } // l + ... + r constexpr ll tri(ll l, ll r) { return (l + r) * (r - l + 1) / 2; } ll max(int x, ll y) { return max((ll)x, y); } ll max(ll x, int y) { return max(x, (ll)y); } int min(int x, ll y) { return min((ll)x, y); } int min(ll x, int y) { return min(x, (ll)y); } // bit 演算系 ll pow2(int i) { return 1LL << i; } int topbit(signed t) { return t == 0 ? -1 : 31 - __builtin_clz(t); } int topbit(ll t) { return t == 0 ? -1 : 63 - __builtin_clzll(t); } int lowbit(signed a) { return a == 0 ? 32 : __builtin_ctz(a); } int lowbit(ll a) { return a == 0 ? 64 : __builtin_ctzll(a); } // int allbit(int n) { return (1 << n) - 1; } constexpr ll mask(int n) { return (1LL << n) - 1; } // int popcount(signed t) { return __builtin_popcount(t); } // int popcount(ll t) { return __builtin_popcountll(t); } int popcount(uint64_t t) { return __builtin_popcountll(t); } static inline uint64_t popcount64(uint64_t x) { uint64_t m1 = 0x5555555555555555ll; uint64_t m2 = 0x3333333333333333ll; uint64_t m4 = 0x0F0F0F0F0F0F0F0Fll; uint64_t h01 = 0x0101010101010101ll; x -= (x >> 1) & m1; x = (x & m2) + ((x >> 2) & m2); x = (x + (x >> 4)) & m4; return (x * h01) >> 56; } bool ispow2(int i) { return i && (i & -i) == i; } ll rnd(ll l, ll r) { //[l, r) #ifdef noimi static mt19937_64 gen; #else static mt19937_64 gen(chrono::steady_clock::now().time_since_epoch().count()); #endif return uniform_int_distribution(l, r - 1)(gen); } ll rnd(ll n) { return rnd(0, n); } template void random_shuffle(vc &a) { rep(i, si(a)) swap(a[i], a[rnd(0, i + 1)]); } int in() { int x; cin >> x; return x; } ll lin() { unsigned long long x; cin >> x; return x; } template pair operator-(const pair &x) { return pair(-x.first, -x.second); } template pair operator-(const pair &x, const pair &y) { return pair(x.fi - y.fi, x.se - y.se); } template pair operator+(const pair &x, const pair &y) { return pair(x.fi + y.fi, x.se + y.se); } template pair operator&(const pair &l, const pair &r) { return pair(max(l.fi, r.fi), min(l.se, r.se)); } template pair operator+=(pair &l, const pair &r) { return l = l + r; } template pair operator-=(pair &l, const pair &r) { return l = l - r; } template bool intersect(const pair &l, const pair &r) { return (l.se < r.se ? r.fi < l.se : l.fi < r.se); } template vector &operator++(vector &v) { fore(e, v) e++; return v; } template vector operator++(vector &v, int) { auto res = v; fore(e, v) e++; return res; } template vector &operator--(vector &v) { fore(e, v) e--; return v; } template vector operator--(vector &v, int) { auto res = v; fore(e, v) e--; return res; } template vector &operator+=(vector &l, const vector &r) { fore(e, r) l.eb(e); return l; } template struct edge { int from, to; T cost; int id; edge(int to, T cost) : from(-1), to(to), cost(cost) {} edge(int from, int to, T cost) : from(from), to(to), cost(cost) {} edge(int from, int to, T cost, int id) : from(from), to(to), cost(cost), id(id) {} constexpr bool operator<(const edge &rhs) const noexcept { return cost < rhs.cost; } edge &operator=(const int &x) { to = x; return *this; } operator int() const { return to; } friend ostream operator<<(ostream &os, edge &e) { return os << e.to; } }; template using Edges = vector>; template Edges read_edges(int m, bool weighted = false) { Edges res; res.reserve(m); for(int i = 0; i < m; i++) { int u, v, c = 0; scan(u), scan(v), u--, v--; if(weighted) scan(c); res.eb(u, v, c, i); } return res; } using Tree = vector>; using Graph = vector>; template using Wgraph = vector>>; Graph getG(int n, int m = -1, bool directed = false, int margin = 1) { Tree res(n); if(m == -1) m = n - 1; while(m--) { int a, b; cin >> a >> b; a -= margin, b -= margin; res[a].emplace_back(b); if(!directed) res[b].emplace_back(a); } return res; } Graph getTreeFromPar(int n, int margin = 1) { Graph res(n); for(int i = 1; i < n; i++) { int a; cin >> a; res[a - margin].emplace_back(i); } return res; } template Wgraph getWg(int n, int m = -1, bool directed = false, int margin = 1) { Wgraph res(n); if(m == -1) m = n - 1; while(m--) { int a, b; T c; scan(a), scan(b), scan(c); a -= margin, b -= margin; res[a].emplace_back(b, c); if(!directed) res[b].emplace_back(a, c); } return res; } void add(Graph &G, int x, int y) { G[x].eb(y), G[y].eb(x); } template void add(Wgraph &G, int x, int y, T c) { G[x].eb(y, c), G[y].eb(x, c); } #define TEST \ INT(testcases); \ while(testcases--) i128 abs(const i128 &x) { return x > 0 ? x : -x; } istream &operator>>(istream &is, i128 &v) { string s; is >> s; v = 0; for(int i = 0; i < (int)s.size(); i++) { if(isdigit(s[i])) { v = v * 10 + s[i] - '0'; } } if(s[0] == '-') { v *= -1; } return is; } ostream &operator<<(ostream &os, const i128 &v) { if(v == 0) { return (os << "0"); } i128 num = v; if(v < 0) { os << '-'; num = -num; } string s; for(; num > 0; num /= 10) { s.push_back((char)(num % 10) + '0'); } reverse(s.begin(), s.end()); return (os << s); } namespace aux { template struct tp { static void output(std::ostream &os, const T &v) { os << std::get(v) << (&os == &cerr ? ", " : " "); tp::output(os, v); } }; template struct tp { static void output(std::ostream &os, const T &v) { os << std::get(v); } }; } // namespace aux template std::ostream &operator<<(std::ostream &os, const std::tuple &t) { if(&os == &cerr) { os << '('; } aux::tp, 0, sizeof...(Ts) - 1>::output(os, t); if(&os == &cerr) { os << ')'; } return os; } template std::ostream &operator<<(std::ostream &os, const priority_queue &_pq) { auto pq = _pq; vector res; while(!empty(pq)) res.emplace_back(pq.top()), pq.pop(); return os << res; } template ostream &operator<<(ostream &os, const pair &p) { if(&os == &cerr) { return os << "(" << p.first << ", " << p.second << ")"; } return os << p.first << " " << p.second; } template std::basic_ostream &operator<<(std::basic_ostream &os, const Container &x) { bool f = true; if(&os == &cerr) os << "["; for(auto &y : x) { if(&os == &cerr) os << (f ? "" : ", ") << y; else os << (f ? "" : " ") << y; f = false; } if(&os == &cerr) os << "]"; return os; } #ifdef noimi #undef endl void debug() { cerr << endl; } void debug(bool) { cerr << endl; } template void debug(bool is_front, Head head, Tail... tail) { if(!is_front) cerr << ", "; cerr << head; debug(false, tail...); } #define dump(args...) \ { \ vector _debug = _split(#args, ','); \ err(true, begin(_debug), args); \ } vector _split(const string &s, char c) { vector v; stringstream ss(s); string x; while(getline(ss, x, c)) { if(empty(v)) v.eb(x); else { bool flag = false; for(auto [c, d] : {pair('(', ')'), pair('[', ']'), pair('{', '}')}) { if(count(all(v.back()), c) != count(all(v.back()), d)) flag = true; } if(flag) v.back() += "," + x; else v.eb(x); } } return move(v); } void err(bool, vector::iterator) { cerr << endl; } template void err(bool is_front, vector::iterator it, T a, Args... args) { if(!is_front) cerr << ", "; cerr << it->substr((*it)[0] == ' ', (*it).size()) << " = " << a, err(false, ++it, args...); } // #define dump(...) cerr << #__VA_ARGS__ << " : ", debug(true, __VA_ARGS__) #else #define dump(...) static_cast(0) #define dbg(...) static_cast(0) #endif void OUT() { cout << endl; } template void OUT(const Head &head, const Tail &...tail) { cout << head; if(sizeof...(tail)) cout << ' '; OUT(tail...); } template static constexpr T inf = numeric_limits::max() / 2; template constexpr pair inf> = {inf, inf}; template struct REC { F f; REC(F &&f_) : f(std::forward(f_)) {} template auto operator()(Args &&...args) const { return f(*this, std::forward(args)...); } }; template vector> runLength(const vector &v) { vector> res; for(auto &e : v) { if(res.empty() or res.back().fi != e) res.eb(e, 1); else res.back().se++; } return res; } vector> runLength(const string &v) { vector> res; for(auto &e : v) { if(res.empty() or res.back().fi != e) res.eb(e, 1); else res.back().se++; } return res; } int toint(const char &c, const char start = 'a') { return c - start; } int toint(const char &c, const string &chars) { return find(all(chars), c) - begin(chars); } int alphabets_to_int(const char &c) { return (islower(c) ? c - 'a' : c - 'A' + 26); } template auto toint(const T &v, const char &start = 'a') { vector ret; ret.reserve(v.size()); for(auto &&e : v) ret.emplace_back(toint(e, start)); return ret; } template auto toint(const T &v, const string &start) { vector ret; ret.reserve(v.size()); for(auto &&e : v) ret.emplace_back(toint(e, start)); return ret; } // a -> 0, A -> 26 template auto alphabets_to_int(const T &s) { vector res; res.reserve(s.size()); for(auto &&e : s) { res.emplace_back(alphabets_to_int(e)); } return res; } template T bin_search(T ok, T ng, const F &f) { while(abs(ok - ng) > 1) { T mid = ok + ng >> 1; (f(mid) ? ok : ng) = mid; } return ok; } template T bin_search_double(T ok, T ng, const F &f, int iter = 80) { while(iter--) { T mid = (ok + ng) / 2; (f(mid) ? ok : ng) = mid; } return ok; } struct Setup_io { Setup_io() { ios_base::sync_with_stdio(0), cin.tie(0), cout.tie(0); cout << fixed << setprecision(11); } } setup_io; #pragma endregion namespace modular { constexpr int MOD = 998244353; const int MAXN = 11000000; template class modint; using mint = modint; using vmint = vector; vector Inv; mint inv(int x); template class modint { public: static constexpr int mod() { return Modulus; } int a; constexpr modint(const ll x = 0) noexcept : a(((x % Modulus) + Modulus) % Modulus) {} constexpr int &val() noexcept { return a; } constexpr const int &val() const noexcept { return a; } constexpr modint operator-() const noexcept { return modint() - *this; } constexpr modint operator+() const noexcept { return *this; } constexpr modint &operator++() noexcept { if(++a == MOD) a = 0; return *this; } constexpr modint &operator--() noexcept { if(!a) a = MOD; a--; return *this; } constexpr modint operator++(int) { modint res = *this; ++*this; return res; } constexpr modint operator--(int) { mint res = *this; --*this; return res; } constexpr modint &operator+=(const modint rhs) noexcept { a += rhs.a; if(a >= Modulus) { a -= Modulus; } return *this; } constexpr modint &operator-=(const modint rhs) noexcept { if(a < rhs.a) { a += Modulus; } a -= rhs.a; return *this; } constexpr modint &operator*=(const modint rhs) noexcept { a = (long long)a * rhs.a % Modulus; return *this; } constexpr modint &operator/=(const modint rhs) noexcept { a = (long long)a * (modular::inv(rhs.a)).a % Modulus; return *this; } constexpr modint pow(long long n) const noexcept { if(n < 0) { n %= Modulus - 1; n = (Modulus - 1) + n; } modint x = *this, r = 1; while(n) { if(n & 1) r *= x; x *= x; n >>= 1; } return r; } constexpr modint inv() const noexcept { return pow(Modulus - 2); } constexpr friend modint operator+(const modint &lhs, const modint &rhs) { return modint(lhs) += modint(rhs); } constexpr friend modint operator-(const modint &lhs, const modint &rhs) { return modint(lhs) -= modint(rhs); } constexpr friend modint operator*(const modint &lhs, const modint &rhs) { return modint(lhs) *= modint(rhs); } constexpr friend modint operator/(const modint &lhs, const modint &rhs) { return modint(lhs) /= modint(rhs); } constexpr friend bool operator==(const modint &lhs, const modint &rhs) { return lhs.a == rhs.a; } constexpr friend bool operator!=(const modint &lhs, const modint &rhs) { return lhs.a != rhs.a; } // constexpr friend modint operator^=(const modint &lhs, const modint &rhs) { return modint(lhs) ^= modint(rhs); } }; vmint Fact{1, 1}, Ifact{1, 1}; mint inv(int n) { if(n > MAXN) return (mint(n)).pow(MOD - 2); if(Inv.empty()) Inv.emplace_back(0), Inv.emplace_back(1); if(Inv.size() > n) return Inv[n]; else { for(int i = Inv.size(); i <= n; ++i) { auto [y, x] = div(int(MOD), i); Inv.emplace_back(Inv[x] * (-y)); } return Inv[n]; } } mint fact(int n) { if(Fact.size() > n) return Fact[n]; else for(int i = Fact.size(); i <= n; ++i) Fact.emplace_back(Fact[i - 1] * i); return Fact[n]; } mint ifact(int n) { if(Ifact.size() > n) return Ifact[n]; else for(int i = Ifact.size(); i <= n; ++i) Ifact.emplace_back(Ifact[i - 1] * inv(i)); return Ifact[n]; } mint modpow(ll a, ll n) { return mint(a).pow(n); } mint inv(mint a) { return inv(a.a); } mint ifact(mint a) { return ifact(a.a); } mint fact(mint a) { return fact(a.a); } mint modpow(mint a, ll n) { return modpow(a.a, n); } mint C(int a, int b) { if(a < 0 || b < 0) return 0; if(a < b) return 0; if(a > MAXN) { mint res = 1; rep(i, b) res *= a - i, res /= i + 1; return res; } return fact(a) * ifact(b) * ifact(a - b); } mint P(int a, int b) { if(a < 0 || b < 0) return 0; if(a < b) return 0; if(a > MAXN) { mint res = 1; rep(i, b) res *= a - i; return res; } return fact(a) * ifact(a - b); } ostream &operator<<(ostream &os, mint a) { os << a.a; return os; } istream &operator>>(istream &is, mint &a) { ll x; is >> x; a = x; return is; } ostream &operator<<(ostream &os, const vmint &a) { if(!a.empty()) { os << a[0]; for(int i = 1; i < si(a); i++) os << " " << a[i]; } return os; } #ifdef _MSC_VER #include #endif namespace convolution { namespace internal { int ceil_pow2(int n) { int x = 0; while((1U << x) < (unsigned int)(n)) x++; return x; } int bsf(unsigned int n) { #ifdef _MSC_VER unsigned long index; _BitScanForward(&index, n); return index; #else return __builtin_ctz(n); #endif } constexpr long long safe_mod(long long x, long long m) { x %= m; if(x < 0) x += m; return x; } struct barrett { unsigned int _m; unsigned long long im; barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {} unsigned int umod() const { return _m; } unsigned int mul(unsigned int a, unsigned int b) const { unsigned long long z = a; z *= b; #ifdef _MSC_VER unsigned long long x; _umul128(z, im, &x); #else unsigned long long x = (unsigned long long)(((unsigned __int128)(z)*im) >> 64); #endif unsigned int v = (unsigned int)(z - x * _m); if(_m <= v) v += _m; return v; } }; constexpr long long pow_mod_constexpr(long long x, long long n, int m) { if(m == 1) return 0; unsigned int _m = (unsigned int)(m); unsigned long long r = 1; unsigned long long y = safe_mod(x, m); while(n) { if(n & 1) r = (r * y) % _m; y = (y * y) % _m; n >>= 1; } return r; } constexpr bool is_prime_constexpr(int n) { if(n <= 1) return false; if(n == 2 || n == 7 || n == 61) return true; if(n % 2 == 0) return false; long long d = n - 1; while(d % 2 == 0) d /= 2; for(long long a : {2, 7, 61}) { long long t = d; long long y = pow_mod_constexpr(a, t, n); while(t != n - 1 && y != 1 && y != n - 1) { y = y * y % n; t <<= 1; } if(y != n - 1 && t % 2 == 0) { return false; } } return true; } template constexpr bool is_prime = is_prime_constexpr(n); constexpr std::pair inv_gcd(long long a, long long b) { a = safe_mod(a, b); if(a == 0) return {b, 0}; long long s = b, t = a; long long m0 = 0, m1 = 1; while(t) { long long u = s / t; s -= t * u; m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } if(m0 < 0) m0 += b / s; return {s, m0}; } // Compile time primitive root // @param m must be prime // @return primitive root (and minimum in now) constexpr int primitive_root_constexpr(int m) { if(m == 2) return 1; if(m == 167772161) return 3; if(m == 469762049) return 3; if(m == 754974721) return 11; if(m == 998244353) return 3; int divs[20] = {}; divs[0] = 2; int cnt = 1; int x = (m - 1) / 2; while(x % 2 == 0) x /= 2; for(int i = 3; (long long)(i)*i <= x; i += 2) { if(x % i == 0) { divs[cnt++] = i; while(x % i == 0) { x /= i; } } } if(x > 1) { divs[cnt++] = x; } for(int g = 2;; g++) { bool ok = true; for(int i = 0; i < cnt; i++) { if(pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) { ok = false; break; } } if(ok) return g; } } template constexpr int primitive_root = primitive_root_constexpr(m); void butterfly(std::vector &a) { static constexpr int g = internal::primitive_root; int n = int(a.size()); int h = internal::ceil_pow2(n); static bool first = true; static mint sum_e[30]; // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i] if(first) { first = false; mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1 int cnt2 = bsf(mint::mod() - 1); mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv(); for(int i = cnt2; i >= 2; i--) { // e^(2^i) == 1 es[i - 2] = e; ies[i - 2] = ie; e *= e; ie *= ie; } mint now = 1; for(int i = 0; i < cnt2 - 2; i++) { sum_e[i] = es[i] * now; now *= ies[i]; } } for(int ph = 1; ph <= h; ph++) { int w = 1 << (ph - 1), p = 1 << (h - ph); mint now = 1; for(int s = 0; s < w; s++) { int offset = s << (h - ph + 1); for(int i = 0; i < p; i++) { auto l = a[i + offset]; auto r = a[i + offset + p] * now; a[i + offset] = l + r; a[i + offset + p] = l - r; } now *= sum_e[bsf(~(unsigned int)(s))]; } } } void butterfly_inv(std::vector &a) { static constexpr int g = internal::primitive_root; int n = int(a.size()); int h = internal::ceil_pow2(n); static bool first = true; static mint sum_ie[30]; // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i] if(first) { first = false; mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1 int cnt2 = bsf(mint::mod() - 1); mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv(); for(int i = cnt2; i >= 2; i--) { // e^(2^i) == 1 es[i - 2] = e; ies[i - 2] = ie; e *= e; ie *= ie; } mint now = 1; for(int i = 0; i < cnt2 - 2; i++) { sum_ie[i] = ies[i] * now; now *= es[i]; } } for(int ph = h; ph >= 1; ph--) { int w = 1 << (ph - 1), p = 1 << (h - ph); mint inow = 1; for(int s = 0; s < w; s++) { int offset = s << (h - ph + 1); for(int i = 0; i < p; i++) { auto l = a[i + offset]; auto r = a[i + offset + p]; a[i + offset] = l + r; a[i + offset + p] = (unsigned long long)(mint::mod() + l.val() - r.val()) * inow.val(); } inow *= sum_ie[bsf(~(unsigned int)(s))]; } } mint z = mint(n).inv(); for(int i = 0; i < n; i++) a[i] *= z; } } // namespace internal std::vector convolution(std::vector a, std::vector b) { int n = int(a.size()), m = int(b.size()); if(!n || !m) return {}; if(std::min(n, m) <= 60) { if(n < m) { std::swap(n, m); std::swap(a, b); } std::vector ans(n + m - 1); for(int i = 0; i < n; i++) { for(int j = 0; j < m; j++) { ans[i + j] += a[i] * b[j]; } } return ans; } int z = 1 << internal::ceil_pow2(n + m - 1); a.resize(z); internal::butterfly(a); b.resize(z); internal::butterfly(b); for(int i = 0; i < z; i++) { a[i] *= b[i]; } internal::butterfly_inv(a); a.resize(n + m - 1); // mint iz = mint(z).inv(); // for(int i = 0; i < n + m - 1; i++) a[i] *= iz; return a; } } // namespace convolution using Poly = vmint; Poly low(const Poly &f, int s) { return Poly(f.begin(), f.begin() + min(max(s, 1), f.size())); } Poly operator-(Poly f) { for(auto &&e : f) e = -e; return f; } Poly &operator+=(Poly &l, const Poly &r) { l.resize(max(l.size(), r.size())); rep(i, r.size()) l[i] += r[i]; return l; } Poly operator+(Poly l, const Poly &r) { return l += r; } Poly &operator-=(Poly &l, const Poly &r) { l.resize(max(l.size(), r.size())); rep(i, r.size()) l[i] -= r[i]; return l; } Poly operator-(Poly l, const Poly &r) { return l -= r; } Poly &operator<<=(Poly &f, size_t n) { return f.insert(f.begin(), n, 0), f; } Poly operator<<(Poly f, size_t n) { return f <<= n; } Poly &operator>>=(Poly &f, size_t n) { return f.erase(f.begin(), f.begin() + min(f.size(), n)), f; } Poly operator>>(Poly f, size_t n) { return f >>= n; } Poly operator*(const Poly &l, const Poly &r) { return convolution::convolution(l, r); } Poly &operator*=(Poly &l, const Poly &r) { return l = l * r; } Poly &operator*=(Poly &l, const mint &x) { for(auto &e : l) e *= x; return l; } Poly operator*(const Poly &l, const mint &x) { auto res = l; return res *= x; } Poly inv(const Poly &f, int s = -1) { if(s == -1) s = f.size(); Poly r(s); r[0] = mint(1) / f[0]; for(int n = 1; n < s; n *= 2) { auto F = low(f, 2 * n); F.resize(2 * n); convolution::internal::butterfly(F); auto g = low(r, 2 * n); g.resize(2 * n); convolution::internal::butterfly(g); rep(i, 2 * n) F[i] *= g[i]; convolution::internal::butterfly_inv(F); rep(i, n) F[i] = 0; convolution::internal::butterfly(F); rep(i, 2 * n) F[i] *= g[i]; convolution::internal::butterfly_inv(F); rep(i, n, min(2 * n, s)) r[i] -= F[i]; } return r; } Poly integ(const Poly &f) { Poly res(f.size() + 1); for(int i = 1; i < (int)res.size(); ++i) res[i] = f[i - 1] / i; return res; } Poly deriv(const Poly &f) { if(f.size() == 0) return Poly(); Poly res(f.size() - 1); rep(i, res.size()) res[i] = f[i + 1] * (i + 1); return res; } Poly log(const Poly &f) { Poly g = integ(inv(f) * deriv(f)); return Poly{g.begin(), g.begin() + f.size()}; } Poly exp(const Poly &f) { Poly g{1}; while(g.size() < f.size()) { Poly x(f.begin(), f.begin() + min(f.size(), g.size() * 2)); x[0] += 1; g.resize(2 * g.size()); x -= log(g); x *= {g.begin(), g.begin() + g.size() / 2}; rep(i, g.size() / 2, min(x.size(), g.size())) g[i] = x[i]; } return {g.begin(), g.begin() + f.size()}; } Poly pow(const Poly &f, ll k, int need = -1) { const int n = (int)f.size(); if(need == -1) need = n; int z = 0; rep(i, n) { if(f[i].a) break; z++; } if(z * k >= need) return Poly(n); mint rev = f[z].inv(); Poly res = exp(log((f >> z) * rev) * k) * f[z].pow(k); res.resize(need - z * k); return res << z * k; } struct Prd { deque deq; Prd() = default; void emplace(const Poly &f) { deq.emplace_back(f); } Poly calc() { if(deq.empty()) return {1}; sort(all(deq), [&](const Poly &f, const Poly &g) { return si(f) < si(g); }); while(deq.size() > 1) { deq.emplace_back(deq[0] * deq[1]); for(int i = 0; i < 2; ++i) deq.pop_front(); } return deq.front(); } }; Poly prd(vector &v) { Prd p; for(auto &e : v) p.emplace(e); return p.calc(); } vmint power_table(mint x, int len) { vmint res(len + 1); res[0] = 1; rep(i, len) res[i + 1] = res[i] * x; return res; } // calc f(x + a) Poly TaylorShift(Poly f, mint a) { int n = f.size(); rep(i, n) f[i] *= fact(i); reverse(all(f)); Poly g(n, 1); rep(i, 1, n) g[i] = g[i - 1] * a * inv(i); f = (f * g); f.resize(n); reverse(begin(f), end(f)); rep(i, n) f[i] *= ifact(i); return f; } // ボールの数、一個以上必要な数、入っていなくてもいい数(区別あり) mint choose(int num, int a, int b = 0) { if(num == 0 and a == 0 and b == 0) return 1; return C(num + b - 1, a + b - 1); } } // namespace modular using namespace modular; namespace segtree_impl { namespace internal { // @param n `0 <= n` // @return minimum non-negative `x` s.t. `n <= 2**x` int ceil_pow2(int n) { int x = 0; while((1U << x) < (unsigned int)(n)) x++; return x; } // @param n `1 <= n` // @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0` constexpr int bsf_constexpr(unsigned int n) { int x = 0; while(!(n & (1 << x))) x++; return x; } // @param n `1 <= n` // @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0` int bsf(unsigned int n) { #ifdef _MSC_VER unsigned long index; _BitScanForward(&index, n); return index; #else return __builtin_ctz(n); #endif } } // namespace internal } // namespace segtree_impl namespace segtree_impl { template struct segtree { public: segtree() : segtree(0) {} explicit segtree(int n) : segtree(std::vector(n, e())) {} explicit segtree(const std::vector &v) : _n(int(v.size())) { log = internal::ceil_pow2(_n); size = 1 << log; d = std::vector(2 * size, e()); for(int i = 0; i < _n; i++) d[size + i] = v[i]; for(int i = size - 1; i >= 1; i--) { update(i); } } void set(int p, S x) { assert(0 <= p && p < _n); p += size; d[p] = x; for(int i = 1; i <= log; i++) update(p >> i); } S get(int p) const { assert(0 <= p && p < _n); return d[p + size]; } S operator[](int k) const { return get(k); } S prod(int l, int r) const { #ifdef ONLINE_JUDGE if(l < 0) l = 0; if(r > _n) r = _n; if(l > r) return e(); #endif assert(0 <= l && l <= r && r <= _n); S sml = e(), smr = e(); l += size; r += size; while(l < r) { if(l & 1) sml = op(sml, d[l++]); if(r & 1) smr = op(d[--r], smr); l >>= 1; r >>= 1; } return op(sml, smr); } S all_prod() const { return d[1]; } template int max_right(int l) const { return max_right(l, [](S x) { return f(x); }); } template int max_right(int l, F f) const { #ifdef ONLINE_JUDGE if(l < 0) l = 0; if(l > _n) l = _n; #endif assert(0 <= l && l <= _n); assert(f(e())); if(l == _n) return _n; l += size; S sm = e(); do { while(l % 2 == 0) l >>= 1; if(!f(op(sm, d[l]))) { while(l < size) { l = (2 * l); if(f(op(sm, d[l]))) { sm = op(sm, d[l]); l++; } } return l - size; } sm = op(sm, d[l]); l++; } while((l & -l) != l); return _n; } template int min_left(int r) const { return min_left(r, [](S x) { return f(x); }); } template int min_left(int r, F f) const { #ifdef ONLINE_JUDGE if(r < 0) r = 0; if(r > _n) r = _n; #endif assert(0 <= r && r <= _n); assert(f(e())); if(r == 0) return 0; r += size; S sm = e(); do { r--; while(r > 1 && (r % 2)) r >>= 1; if(!f(op(d[r], sm))) { while(r < size) { r = (2 * r + 1); if(f(op(d[r], sm))) { sm = op(d[r], sm); r--; } } return r + 1 - size; } sm = op(d[r], sm); } while((r & -r) != r); return 0; } friend std::ostream &operator<<(std::ostream &os, segtree &ls) { os << "{"; for(int i = 0; i < ls._n; i++) os << ls.get(i) << (i == ls._n - 1 ? "" : ", "); return os << "}"; } private: int _n, size, log; std::vector d; void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); } }; template constexpr T inf = std::numeric_limits::max() / 2; template constexpr std::pair inf> = {inf, inf}; template T min_func(T x, T y) { return min(x, y); } template T max_func(T x, T y) { return max(x, y); } template T plus_func(T x, T y) { return x + y; } template T set_func(T x, T y) { return (y == ID ? x : y); } template T min_e() { return inf; } template T max_e() { return -inf; } template T plus_e() { return T(); } template using RST = segtree, plus_e>; template using RmT = segtree, min_e>; template using RMT = segtree, max_e>; } // namespace segtree_impl using segtree_impl::RmT; using segtree_impl::RMT; using segtree_impl::RST; using segtree_impl::segtree; int main() { STR(s); auto a = toint(s, 'a'); VEC(int, b, 3); vi cnt(3); rep(i, 3) cnt[i] = count(all(a), i); rep(i, 3) b[i] -= cnt[i]; mint ans; vmint fz(b[0] + b[1] + 1), fx(b[0] + 1), fy(b[1] + 1); rep(z, si(fz)) fz[z] = choose(b[2], 0, 1 + cnt[0] + cnt[1] + b[0] + b[1] - z) * choose(z, 0, cnt[2]); rep(x, si(fx)) fx[x] = ifact(x) * choose(b[0] - x, 0, cnt[1]); rep(y, si(fy)) fy[y] = ifact(y) * choose(b[1] - y, 0, cnt[0]); auto f = fx * fy; auto FZ = RUI(fz); // dump(f, fx, fy, fz, FZ); rep(i, si(f)) { ans += fact(i) * f[i] * FZ[i + 1]; } OUT(ans); // fz[i] = C(cnt[0] + cnt[1] + b[0] + b[1] - i + b[2], b[2]) * C(z + cnt) // rep(x, b[0] + 1) rep(y, b[1] + 1) rep(z, x + y + 1) { // // ans += C(x + y, x) * C(cnt[0] + cnt[1] + b[0] + b[1] - z + b[2], b[2]) * C(z + cnt[2] - 1, cnt[2] - 1) * C(b[0] - x + cnt[1] - 1, cnt[1] - 1) * // // C(b[1] - y + cnt[0] - 1, cnt[0] - 1); // ans += C(x + y, x) * choose(b[2], 0, 1 + cnt[0] + cnt[1] + b[0] + b[1] - z) * choose(z, 0, cnt[2]) * choose(b[0] - x, 0, cnt[1]) * // choose(b[1] - y, 0, cnt[0]); // } // OUT(ans); }