#line 1 "A.cpp" // #pragma GCC target("avx2") // #pragma GCC optimize("O3") // #pragma GCC optimize("unroll-loops") #include using namespace std; using ll = long long; using ull = unsigned long long; template using pq = priority_queue; template using qp = priority_queue, greater>; #define vec(T, A, ...) vector A(__VA_ARGS__); #define vvec(T, A, h, ...) vector> A(h, vector(__VA_ARGS__)); #define vvvec(T, A, h1, h2, ...) vector>> A(h1, vector>(h2, vector(__VA_ARGS__))); #ifndef RIN__LOCAL #define endl "\n" #endif #define spa ' ' #define len(A) A.size() #define all(A) begin(A), end(A) #define fori1(a) for(ll _ = 0; _ < (a); _++) #define fori2(i, a) for(ll i = 0; i < (a); i++) #define fori3(i, a, b) for(ll i = (a); i < (b); i++) #define fori4(i, a, b, c) for(ll i = (a); ((c) > 0 || i > (b)) && ((c) < 0 || i < (b)); i += (c)) #define overload4(a, b, c, d, e, ...) e #define fori(...) overload4(__VA_ARGS__, fori4, fori3, fori2, fori1)(__VA_ARGS__) template vector> ENUMERATE(vector &A, ll s = 0){ vector> ret(A.size()); for(int i = 0; i < A.size(); i++) ret[i] = {i + s, A[i]}; return ret; } vector> ENUMERATE(string &A, ll s = 0){ vector> ret(A.size()); for(int i = 0; i < A.size(); i++) ret[i] = {i + s, A[i]}; return ret; } #define enum1(A) fori(A.size()) #define enum2(a, A) for(auto a:A) #define enum3(i, a, A) for(auto&& [i, a]: ENUMERATE(A)) #define enum4(i, a, A, s) for(auto&& [i, a]: ENUMERATE(A, s)) #define enum(...) overload4(__VA_ARGS__, enum4, enum3, enum2, enum1)(__VA_ARGS__) template vector> ZIP(vector &A, vector &B){ int n = min(A.size(), B.size()); vector> ret(n); for(int i = 0; i < n; i++) ret[i] = {A[i], B[i]}; return ret; } template vector> ENUMZIP(vector &A, vector &B, ll s = 0){ int n = min(A.size(), B.size()); vector> ret(n); for(int i = 0; i < n; i++) ret[i] = {i + s, A[i], B[i]}; return ret; } #define zip4(a, b, A, B) for(auto&& [a, b]: ZIP(A, B)) #define enumzip5(i, a, b, A, B) for(auto&& [i, a, b]: ENUMZIP(A, B)) #define enumzip6(i, a, b, A, B, s) for(auto&& [i, a, b]: ENUMZIP(A, B, s)) #define overload6(a, b, c, d, e, f, g, ...) g #define zip(...) overload6(__VA_ARGS__, enumzip6, enumzip5, zip4, _, _, _)(__VA_ARGS__) vector stoc(string &S){ int n = S.size(); vector ret(n); for(int i = 0; i < n; i++) ret[i] = S[i]; return ret; } #define INT(...) int __VA_ARGS__; inp(__VA_ARGS__); #define LL(...) ll __VA_ARGS__; inp(__VA_ARGS__); #define STRING(...) string __VA_ARGS__; inp(__VA_ARGS__); #define CHAR(...) char __VA_ARGS__; inp(__VA_ARGS__); #define VEC(T, A, n) vector A(n); inp(A); #define VVEC(T, A, n, m) vector> A(n, vector(m)); inp(A); const ll MOD1 = 1000000007; const ll MOD9 = 998244353; template auto min(const T& a){ return *min_element(all(a)); } template auto max(const T& a){ return *max_element(all(a)); } 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); } void FLUSH(){cout << flush;} void print(){cout << endl;} template void print(Head &&head, Tail &&... tail) { cout << head; if (sizeof...(Tail)) cout << spa; print(forward(tail)...); } template void print(vector &A){ int n = A.size(); for(int i = 0; i < n; i++){ cout << A[i]; if(i != n - 1) cout << ' '; } cout << endl; } template void print(vector> &A){ for(auto &row: A) print(row); } template void print(pair &A){ cout << A.first << spa << A.second << endl; } template void print(vector> &A){ for(auto &row: A) print(row); } template void prisep(vector &A, S sep){ int n = A.size(); for(int i = 0; i < n; i++){ cout << A[i]; if(i == n - 1) cout << endl; else cout << sep; } } template void priend(T A, S end){ cout << A << end; } template void priend(T A){ priend(A, spa); } template void inp(T&... a){ (cin >> ... >> a); } template void inp(vector &A){ for(auto &a:A) cin >> a; } template void inp(vector> &A){ for(auto &row:A) inp(row); } template void inp(pair &A){ inp(A.first, A.second); } template void inp(vector> &A){ for(auto &row: A) inp(row.first, row.second); } template T sum(vector &A){ T tot = 0; for(auto a:A) tot += a; return tot; } template pair, map> compression(vector X){ sort(all(X)); X.erase(unique(all(X)), X.end()); map mp; for(int i = 0; i < X.size(); i++) mp[X[i]] = i; return {X, mp}; } #line 1 "atcoder/convolution.hpp" #line 7 "atcoder/convolution.hpp" #include #line 9 "atcoder/convolution.hpp" #line 1 "atcoder/internal_bit.hpp" #ifdef _MSC_VER #include #endif namespace atcoder { 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 atcoder #line 1 "atcoder/modint.hpp" #line 7 "atcoder/modint.hpp" #ifdef _MSC_VER #include #endif #line 1 "atcoder/internal_math.hpp" #line 5 "atcoder/internal_math.hpp" #ifdef _MSC_VER #include #endif namespace atcoder { namespace internal { // @param m `1 <= m` // @return x mod m constexpr long long safe_mod(long long x, long long m) { x %= m; if (x < 0) x += m; return x; } // Fast modular multiplication by barrett reduction // Reference: https://en.wikipedia.org/wiki/Barrett_reduction // NOTE: reconsider after Ice Lake struct barrett { unsigned int _m; unsigned long long im; // @param m `1 <= m < 2^31` explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {} // @return m unsigned int umod() const { return _m; } // @param a `0 <= a < m` // @param b `0 <= b < m` // @return `a * b % m` unsigned int mul(unsigned int a, unsigned int b) const { // [1] m = 1 // a = b = im = 0, so okay // [2] m >= 2 // im = ceil(2^64 / m) // -> im * m = 2^64 + r (0 <= r < m) // let z = a*b = c*m + d (0 <= c, d < m) // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2 // ((ab * im) >> 64) == c or c + 1 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; } }; // @param n `0 <= n` // @param m `1 <= m` // @return `(x ** n) % m` 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; } // Reference: // M. Forisek and J. Jancina, // Fast Primality Testing for Integers That Fit into a Machine Word // @param n `0 <= n` 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; constexpr long long bases[3] = {2, 7, 61}; for (long long a : bases) { 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); // @param b `1 <= b` // @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g constexpr std::pair inv_gcd(long long a, long long b) { a = safe_mod(a, b); if (a == 0) return {b, 0}; // Contracts: // [1] s - m0 * a = 0 (mod b) // [2] t - m1 * a = 0 (mod b) // [3] s * |m1| + t * |m0| <= b 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 // [3]: // (s - t * u) * |m1| + t * |m0 - m1 * u| // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u) // = s * |m1| + t * |m0| <= b auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } // by [3]: |m0| <= b/g // by g != b: |m0| < b/g 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); // @param n `n < 2^32` // @param m `1 <= m < 2^32` // @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64) unsigned long long floor_sum_unsigned(unsigned long long n, unsigned long long m, unsigned long long a, unsigned long long b) { unsigned long long ans = 0; while (true) { if (a >= m) { ans += n * (n - 1) / 2 * (a / m); a %= m; } if (b >= m) { ans += n * (b / m); b %= m; } unsigned long long y_max = a * n + b; if (y_max < m) break; // y_max < m * (n + 1) // floor(y_max / m) <= n n = (unsigned long long)(y_max / m); b = (unsigned long long)(y_max % m); std::swap(m, a); } return ans; } } // namespace internal } // namespace atcoder #line 1 "atcoder/internal_type_traits.hpp" #line 7 "atcoder/internal_type_traits.hpp" namespace atcoder { namespace internal { #ifndef _MSC_VER template using is_signed_int128 = typename std::conditional::value || std::is_same::value, std::true_type, std::false_type>::type; template using is_unsigned_int128 = typename std::conditional::value || std::is_same::value, std::true_type, std::false_type>::type; template using make_unsigned_int128 = typename std::conditional::value, __uint128_t, unsigned __int128>; template using is_integral = typename std::conditional::value || is_signed_int128::value || is_unsigned_int128::value, std::true_type, std::false_type>::type; template using is_signed_int = typename std::conditional<(is_integral::value && std::is_signed::value) || is_signed_int128::value, std::true_type, std::false_type>::type; template using is_unsigned_int = typename std::conditional<(is_integral::value && std::is_unsigned::value) || is_unsigned_int128::value, std::true_type, std::false_type>::type; template using to_unsigned = typename std::conditional< is_signed_int128::value, make_unsigned_int128, typename std::conditional::value, std::make_unsigned, std::common_type>::type>::type; #else template using is_integral = typename std::is_integral; template using is_signed_int = typename std::conditional::value && std::is_signed::value, std::true_type, std::false_type>::type; template using is_unsigned_int = typename std::conditional::value && std::is_unsigned::value, std::true_type, std::false_type>::type; template using to_unsigned = typename std::conditional::value, std::make_unsigned, std::common_type>::type; #endif template using is_signed_int_t = std::enable_if_t::value>; template using is_unsigned_int_t = std::enable_if_t::value>; template using to_unsigned_t = typename to_unsigned::type; } // namespace internal } // namespace atcoder #line 14 "atcoder/modint.hpp" namespace atcoder { namespace internal { struct modint_base {}; struct static_modint_base : modint_base {}; template using is_modint = std::is_base_of; template using is_modint_t = std::enable_if_t::value>; } // namespace internal template * = nullptr> struct static_modint : internal::static_modint_base { using mint = static_modint; public: static constexpr int mod() { return m; } static mint raw(int v) { mint x; x._v = v; return x; } static_modint() : _v(0) {} template * = nullptr> static_modint(T v) { long long x = (long long)(v % (long long)(umod())); if (x < 0) x += umod(); _v = (unsigned int)(x); } template * = nullptr> static_modint(T v) { _v = (unsigned int)(v % umod()); } unsigned int val() const { return _v; } mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator-=(const mint& rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } mint& operator*=(const mint& rhs) { unsigned long long z = _v; z *= rhs._v; _v = (unsigned int)(z % umod()); return *this; } mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } mint pow(long long n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } mint inv() const { if (prime) { assert(_v); return pow(umod() - 2); } else { auto eg = internal::inv_gcd(_v, m); assert(eg.first == 1); return eg.second; } } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } private: unsigned int _v; static constexpr unsigned int umod() { return m; } static constexpr bool prime = internal::is_prime; }; template struct dynamic_modint : internal::modint_base { using mint = dynamic_modint; public: static int mod() { return (int)(bt.umod()); } static void set_mod(int m) { assert(1 <= m); bt = internal::barrett(m); } static mint raw(int v) { mint x; x._v = v; return x; } dynamic_modint() : _v(0) {} template * = nullptr> dynamic_modint(T v) { long long x = (long long)(v % (long long)(mod())); if (x < 0) x += mod(); _v = (unsigned int)(x); } template * = nullptr> dynamic_modint(T v) { _v = (unsigned int)(v % mod()); } unsigned int val() const { return _v; } mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator-=(const mint& rhs) { _v += mod() - rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator*=(const mint& rhs) { _v = bt.mul(_v, rhs._v); return *this; } mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } mint pow(long long n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } mint inv() const { auto eg = internal::inv_gcd(_v, mod()); assert(eg.first == 1); return eg.second; } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } private: unsigned int _v; static internal::barrett bt; static unsigned int umod() { return bt.umod(); } }; template internal::barrett dynamic_modint::bt(998244353); using modint998244353 = static_modint<998244353>; using modint1000000007 = static_modint<1000000007>; using modint = dynamic_modint<-1>; namespace internal { template using is_static_modint = std::is_base_of; template using is_static_modint_t = std::enable_if_t::value>; template struct is_dynamic_modint : public std::false_type {}; template struct is_dynamic_modint> : public std::true_type {}; template using is_dynamic_modint_t = std::enable_if_t::value>; } // namespace internal } // namespace atcoder #line 12 "atcoder/convolution.hpp" namespace atcoder { namespace internal { template , internal::is_static_modint_t* = nullptr> struct fft_info { static constexpr int rank2 = bsf_constexpr(mint::mod() - 1); std::array root; // root[i]^(2^i) == 1 std::array iroot; // root[i] * iroot[i] == 1 std::array rate2; std::array irate2; std::array rate3; std::array irate3; fft_info() { root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2); iroot[rank2] = root[rank2].inv(); for (int i = rank2 - 1; i >= 0; i--) { root[i] = root[i + 1] * root[i + 1]; iroot[i] = iroot[i + 1] * iroot[i + 1]; } { mint prod = 1, iprod = 1; for (int i = 0; i <= rank2 - 2; i++) { rate2[i] = root[i + 2] * prod; irate2[i] = iroot[i + 2] * iprod; prod *= iroot[i + 2]; iprod *= root[i + 2]; } } { mint prod = 1, iprod = 1; for (int i = 0; i <= rank2 - 3; i++) { rate3[i] = root[i + 3] * prod; irate3[i] = iroot[i + 3] * iprod; prod *= iroot[i + 3]; iprod *= root[i + 3]; } } } }; template * = nullptr> void butterfly(std::vector& a) { int n = int(a.size()); int h = internal::ceil_pow2(n); static const fft_info info; int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed while (len < h) { if (h - len == 1) { int p = 1 << (h - len - 1); mint rot = 1; for (int s = 0; s < (1 << len); s++) { int offset = s << (h - len); for (int i = 0; i < p; i++) { auto l = a[i + offset]; auto r = a[i + offset + p] * rot; a[i + offset] = l + r; a[i + offset + p] = l - r; } if (s + 1 != (1 << len)) rot *= info.rate2[bsf(~(unsigned int)(s))]; } len++; } else { // 4-base int p = 1 << (h - len - 2); mint rot = 1, imag = info.root[2]; for (int s = 0; s < (1 << len); s++) { mint rot2 = rot * rot; mint rot3 = rot2 * rot; int offset = s << (h - len); for (int i = 0; i < p; i++) { auto mod2 = 1ULL * mint::mod() * mint::mod(); auto a0 = 1ULL * a[i + offset].val(); auto a1 = 1ULL * a[i + offset + p].val() * rot.val(); auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val(); auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val(); auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val(); auto na2 = mod2 - a2; a[i + offset] = a0 + a2 + a1 + a3; a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3)); a[i + offset + 2 * p] = a0 + na2 + a1na3imag; a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag); } if (s + 1 != (1 << len)) rot *= info.rate3[bsf(~(unsigned int)(s))]; } len += 2; } } } template * = nullptr> void butterfly_inv(std::vector& a) { int n = int(a.size()); int h = internal::ceil_pow2(n); static const fft_info info; int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed while (len) { if (len == 1) { int p = 1 << (h - len); mint irot = 1; for (int s = 0; s < (1 << (len - 1)); s++) { int offset = s << (h - len + 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()) * irot.val(); ; } if (s + 1 != (1 << (len - 1))) irot *= info.irate2[bsf(~(unsigned int)(s))]; } len--; } else { // 4-base int p = 1 << (h - len); mint irot = 1, iimag = info.iroot[2]; for (int s = 0; s < (1 << (len - 2)); s++) { mint irot2 = irot * irot; mint irot3 = irot2 * irot; int offset = s << (h - len + 2); for (int i = 0; i < p; i++) { auto a0 = 1ULL * a[i + offset + 0 * p].val(); auto a1 = 1ULL * a[i + offset + 1 * p].val(); auto a2 = 1ULL * a[i + offset + 2 * p].val(); auto a3 = 1ULL * a[i + offset + 3 * p].val(); auto a2na3iimag = 1ULL * mint((mint::mod() + a2 - a3) * iimag.val()).val(); a[i + offset] = a0 + a1 + a2 + a3; a[i + offset + 1 * p] = (a0 + (mint::mod() - a1) + a2na3iimag) * irot.val(); a[i + offset + 2 * p] = (a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) * irot2.val(); a[i + offset + 3 * p] = (a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) * irot3.val(); } if (s + 1 != (1 << (len - 2))) irot *= info.irate3[bsf(~(unsigned int)(s))]; } len -= 2; } } } template * = nullptr> std::vector convolution_naive(const std::vector& a, const std::vector& b) { int n = int(a.size()), m = int(b.size()); std::vector ans(n + m - 1); if (n < m) { for (int j = 0; j < m; j++) { for (int i = 0; i < n; i++) { ans[i + j] += a[i] * b[j]; } } } else { for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { ans[i + j] += a[i] * b[j]; } } } return ans; } template * = nullptr> std::vector convolution_fft(std::vector a, std::vector b) { int n = int(a.size()), m = int(b.size()); 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 internal template * = nullptr> 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) return convolution_naive(a, b); return internal::convolution_fft(a, b); } template * = nullptr> std::vector convolution(const std::vector& a, const std::vector& b) { int n = int(a.size()), m = int(b.size()); if (!n || !m) return {}; if (std::min(n, m) <= 60) return convolution_naive(a, b); return internal::convolution_fft(a, b); } template ::value>* = nullptr> std::vector convolution(const std::vector& a, const std::vector& b) { int n = int(a.size()), m = int(b.size()); if (!n || !m) return {}; using mint = static_modint; std::vector a2(n), b2(m); for (int i = 0; i < n; i++) { a2[i] = mint(a[i]); } for (int i = 0; i < m; i++) { b2[i] = mint(b[i]); } auto c2 = convolution(move(a2), move(b2)); std::vector c(n + m - 1); for (int i = 0; i < n + m - 1; i++) { c[i] = c2[i].val(); } return c; } std::vector convolution_ll(const std::vector& a, const std::vector& b) { int n = int(a.size()), m = int(b.size()); if (!n || !m) return {}; static constexpr unsigned long long MOD1 = 754974721; // 2^24 static constexpr unsigned long long MOD2 = 167772161; // 2^25 static constexpr unsigned long long MOD3 = 469762049; // 2^26 static constexpr unsigned long long M2M3 = MOD2 * MOD3; static constexpr unsigned long long M1M3 = MOD1 * MOD3; static constexpr unsigned long long M1M2 = MOD1 * MOD2; static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3; static constexpr unsigned long long i1 = internal::inv_gcd(MOD2 * MOD3, MOD1).second; static constexpr unsigned long long i2 = internal::inv_gcd(MOD1 * MOD3, MOD2).second; static constexpr unsigned long long i3 = internal::inv_gcd(MOD1 * MOD2, MOD3).second; auto c1 = convolution(a, b); auto c2 = convolution(a, b); auto c3 = convolution(a, b); std::vector c(n + m - 1); for (int i = 0; i < n + m - 1; i++) { unsigned long long x = 0; x += (c1[i] * i1) % MOD1 * M2M3; x += (c2[i] * i2) % MOD2 * M1M3; x += (c3[i] * i3) % MOD3 * M1M2; // B = 2^63, -B <= x, r(real value) < B // (x, x - M, x - 2M, or x - 3M) = r (mod 2B) // r = c1[i] (mod MOD1) // focus on MOD1 // r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B) // r = x, // x - M' + (0 or 2B), // x - 2M' + (0, 2B or 4B), // x - 3M' + (0, 2B, 4B or 6B) (without mod!) // (r - x) = 0, (0) // - M' + (0 or 2B), (1) // -2M' + (0 or 2B or 4B), (2) // -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1) // we checked that // ((1) mod MOD1) mod 5 = 2 // ((2) mod MOD1) mod 5 = 3 // ((3) mod MOD1) mod 5 = 4 long long diff = c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1)); if (diff < 0) diff += MOD1; static constexpr unsigned long long offset[5] = { 0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3}; x -= offset[diff % 5]; c[i] = x; } return c; } } // namespace atcoder #line 3 "_tmp/wildcard_matching.hpp" template vector wildcard_matching(vector &S, vector &T){ int n = S.size(); int m = T.size(); vector S3(n); vector S2(n); vector S1(n); for(int i = 0; i < n; i++){ long long x = S[i]; long long y = int(S[i] > 0); S3[i] = y * x * x % MOD; S2[i] = y * x; S1[i] = y; } vector T3(n); vector T2(n); vector T1(n); for(int i = 0; i < m; i++){ long long x = T[i]; long long y = int(T[i] > 0); T3[m - 1 - i] = y * x * x % MOD; T2[m - 1 - i] = y * x; T1[m - 1 - i] = y; } auto res1 = atcoder::convolution(S3, T1); auto res2 = atcoder::convolution(S2, T2); auto res3 = atcoder::convolution(S1, T3); vector res(n - m + 1); for(int i = 0; i < n - m + 1; i++){ long long x = res1[i + m - 1] - 2 * res2[i + m - 1] + res3[i + m - 1]; res[i] = bool(x == 0); } return res; } vector wildcard_matching(vector &S, vector &T){ auto res1 = wildcard_matching<998244353>(S, T); auto res2 = wildcard_matching<924844033>(S, T); auto res3 = wildcard_matching<1012924417>(S, T); vector res; for(int i = 0; i < res1.size(); i++){ if(res1[i] && res2[i] && res3[i]) res.push_back(i); } return res; } template vector wildcard_matching(vector &S, vector &T){ int n = S.size(); int m = T.size(); vector S3(n); vector S2(n); vector S1(n); for(int i = 0; i < n; i++){ long long x = S[i]; long long y = int(S[i] > 0); S3[i] = y * x * x % MOD; S2[i] = y * x; S1[i] = y; } vector T3(n); vector T2(n); vector T1(n); for(int i = 0; i < m; i++){ long long x = T[i]; long long y = int(T[i] > 0); T3[m - 1 - i] = y * x * x % MOD; T2[m - 1 - i] = y * x; T1[m - 1 - i] = y; } auto res1 = atcoder::convolution(S3, T1); auto res2 = atcoder::convolution(S2, T2); auto res3 = atcoder::convolution(S1, T3); vector res(n - m + 1); for(int i = 0; i < n - m + 1; i++){ long long x = res1[i + m - 1] - 2 * res2[i + m - 1] + res3[i + m - 1]; res[i] = bool(x == 0); } return res; } vector wildcard_matching(vector &S, vector &T){ auto res1 = wildcard_matching<998244353>(S, T); auto res2 = wildcard_matching<924844033>(S, T); auto res3 = wildcard_matching<1012924417>(S, T); vector res; for(int i = 0; i < res1.size(); i++){ if(res1[i] && res2[i] && res3[i]) res.push_back(i); } return res; } vector wildcard_matching(vector &S, vector &T, char wild='?'){ char mi = S[0]; int n = S.size(); int m = T.size(); for(int i = 0; i < n; i++){ if(S[i] != wild && S[i] < mi) mi = S[i]; } for(int i = 0; i < m; i++){ if(T[i] != wild && T[i] < mi) mi = T[i]; } vector SS(n); vector TT(m); for(int i = 0; i < n; i++){ if(S[i] == wild) SS[i] = 0; else SS[i] = S[i] - mi + 1; } for(int i = 0; i < m; i++){ if(T[i] == wild) TT[i] = 0; else TT[i] = T[i] - mi + 1; } auto res1 = wildcard_matching<998244353>(SS, TT); auto res2 = wildcard_matching<924844033>(SS, TT); auto res3 = wildcard_matching<1012924417>(SS, TT); vector res; for(int i = 0; i < res1.size(); i++){ if(res1[i] && res2[i] && res3[i]) res.push_back(i); } return res; } vector wildcard_matching(string &S, string &T, char wild='?'){ char mi = S[0]; int n = S.size(); int m = T.size(); for(int i = 0; i < n; i++){ if(S[i] != wild && S[i] < mi) mi = S[i]; } for(int i = 0; i < m; i++){ if(T[i] != wild && T[i] < mi) mi = T[i]; } vector SS(n); vector TT(m); for(int i = 0; i < n; i++){ if(S[i] == wild) SS[i] = 0; else SS[i] = S[i] - mi + 1; } for(int i = 0; i < m; i++){ if(T[i] == wild) TT[i] = 0; else TT[i] = T[i] - mi + 1; } auto res1 = wildcard_matching<998244353>(SS, TT); auto res2 = wildcard_matching<924844033>(SS, TT); auto res3 = wildcard_matching<1012924417>(SS, TT); vector res; for(int i = 0; i < res1.size(); i++){ if(res1[i] && res2[i] && res3[i]) res.push_back(i); } return res; } #line 2 "Library/C++/string/safetyRollingHash.hpp" template struct RollingHash{ using u64 = uint64_t; using u128 = __uint128_t; int n; u64 base; const u64 MOD = (1ull << 61ull) - 1; vector pw, h; RollingHash(vector &S, u64 base) : base(base){ n = S.size(); pw.assign(n + 1, 1ull); h.assign(n + 1, 0ull); for(int i = 0; i < n; i++){ pw[i + 1] = Mul(pw[i], base); h[i + 1] = Add(Mul(h[i], base), S[i]); } } RollingHash(string &S, u64 base) : base(base){ n = S.size(); pw.assign(n + 1, 1ull); h.assign(n + 1, 0ull); for(int i = 0; i < n; i++){ pw[i + 1] = Mul(pw[i], base); h[i + 1] = Add(Mul(h[i], base), S[i]); } } u64 get(int l, int r){ return Add(h[r], MOD - Mul(h[l], pw[r - l])); } u64 Mul(u64 a, u64 b){ u128 c = (u128) a * b; return Add(c >> 61, c & MOD); } u64 Add(u64 a, u64 b){ a += b; if(a >= MOD) a -= MOD; return a; } }; #line 2 "Library/C++/other/RandomNumberGenerator.hpp" struct RandomNumberGenerator{ mt19937 mt; RandomNumberGenerator() : mt(chrono::steady_clock::now().time_since_epoch().count()) {} int operator()(int a, int b){ uniform_int_distribution< int > dist(a, b - 1); return dist(mt); } int operator()(int b){ return (*this)(0, b); } }; #line 190 "A.cpp" void solve(){ INT(n); STRING(S); string T = "helloworld"; int m = T.size(); auto res = wildcard_matching(S, T); if(res.empty()){ print(-1); return; } fori(i, n){ if(S[i] == '?') S[i] = 'a'; } RandomNumberGenerator rnd; int base = rnd(1 << 30); RollingHash rhs(S, base); RollingHash rht(T, base); auto calc=[&](int i, int l){ if(l <= i){ return rhs.get(0, l); } else if(l <= i + m){ return rhs.Add(rhs.Mul(rhs.get(0, i), rhs.pw[l - i]), rht.get(0, l - i)); } else{ auto res = rhs.Add(rhs.Mul(rhs.get(0, i), rhs.pw[m]), rht.get(0, m)); return rhs.Add(res, rhs.Mul(res, rhs.pw[l - i - m])), rhs.get(i + m, l); } }; auto get=[&](int i, int l){ if(l < i){ return S[l]; } else if(l < i + m){ return T[l - i]; } else{ return S[l]; } }; int ind = res[0]; fori(i, 1, res.size()){ int l = 0; int r = n; while(r - l > 1){ int mid = (l + r) / 2; if(calc(res[i], mid) == calc(ind, mid)) l = mid; else r = mid; } if(get(res[i], l) < get(ind, l)) ind = res[i]; } fori(i, m){ S[i + ind] = T[i]; } print(S); } int main(){ cin.tie(0)->sync_with_stdio(0); // cout << fixed << setprecision(12); int t; t = 1; cin >> t; while(t--) solve(); return 0; }