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\n#pragma GCC optimize(\"Ofast\")\nusing namespace std;\n\n//#define ACL\n#ifdef ACL\n#if __has_include()\n#include \nusing namespace atcoder;\n#endif\n#if __has_include(\"acl-all.h\")\n#include \"acl-all.h\" //on Wandbox\nusing namespace atcoder;\n#endif\n#endif //#ifdef ACL\n\ntypedef long double ld;\ntypedef long long int ll;\ntypedef unsigned long long int ull;\ntypedef vector vi;\ntypedef vector vc;\ntypedef vector vb;\ntypedef vector vd;\ntypedef vector vs;\ntypedef vector vll;\ntypedef vector > vpll;\ntypedef vector > vvi;\ntypedef vector > vvc;\ntypedef vector > vvs;\ntypedef vector > vvll;\ntypedef vector > vvb;\ntypedef pair pll;\n\n#define rep(i, n) for (int i = 0; i < int(n); ++i)\n#define rrep(i, n) for (int i = 1; i <= int(n); ++i)\n#define irep(it, stl) for (auto it = stl.begin(); it != stl.end(); it++)\n#define drep(i, n) for (int i = int(n)-1; i >= 0; --i)\n\n#define MES(a) MES2 a\n#define MES2(a0,a1,a2,a3,a4,x,...) x\n#define mes_1(x1) cout< inline bool chmax(T& a, T b) { if (a < b) { a = b; return 1; } return 0; }\ntemplate inline bool chmin(T& a, T b) { if (a > b) { a = b; return 1; } return 0; }\nstruct io { io() { ios::sync_with_stdio(false); cin.tie(0); } };\nconst int INF = INT_MAX;\nconst ll LLINF = 1LL << 60;\nconst ll MOD = 1000000007;\nconst double EPS = 1e-9;\n\nll fact_mod(ll n, ll mod) {\n ll f = 1; for (ll i = 2; i <= n; i++) f = f * (i % mod) % mod;\n return f;\n}\n\nll modpow(ll x, ll n, ll mod) {\n if(n == 0) return 1;\n ll res = modpow((x * x) % mod, n / 2 , mod);\n if(n & 1) res = (res * x) % mod;\n return res;\n}\n\nll modncr(ll n, ll r, ll mod) {\n if(r > n-r) r = n-r;\n if(r == 0) return 1;\n ll a = 1;\n rep(i, r) a = a * ((n-i) % mod) % mod;\n ll b = modpow(fact_mod(r, mod), mod-2, mod);\n return (a % mod) * (b % mod) % mod;\n}\n\n\nsigned main() {\n\t\n\t\n\t\n\t\n}\n","codes",[3388],{"_3389":3390,"_3384":3391},"file","acl-all.h","//ACL-ALL.H\n\n#ifndef ATCODER_INTERNAL_BITOP_HPP\n#define ATCODER_INTERNAL_BITOP_HPP 1\n#ifdef _MSC_VER\n#include \n#endif\nnamespace atcoder {\nnamespace internal {\n// @param n `0 <= n`\n// @return minimum non-negative `x` s.t. `n <= 2**x`\nint ceil_pow2(int n) {\n int x = 0;\n while ((1U << x) < (unsigned int)(n)) x++;\n return x;\n}\n// @param n `1 <= n`\n// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`\nint bsf(unsigned int n) {\n#ifdef _MSC_VER\n unsigned long index;\n _BitScanForward(&index, n);\n return index;\n#else\n return __builtin_ctz(n);\n#endif\n}\n} // namespace internal\n} // namespace atcoder\n#endif // ATCODER_INTERNAL_BITOP_HPP\n#ifndef ATCODER_INTERNAL_MATH_HPP\n#define ATCODER_INTERNAL_MATH_HPP 1\n#include \nnamespace atcoder {\nnamespace internal {\n// @param m `1 <= m`\n// @return x mod m\nconstexpr long long safe_mod(long long x, long long m) {\n x %= m;\n if (x < 0) x += m;\n return x;\n}\n// Fast moduler by barrett reduction\n// Reference: https://en.wikipedia.org/wiki/Barrett_reduction\n// NOTE: reconsider after Ice Lake\nstruct barrett {\n unsigned int _m;\n unsigned long long im;\n // @param m `1 <= m`\n barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}\n // @return m\n unsigned int umod() const { return _m; }\n // @param a `0 <= a < m`\n // @param b `0 <= b < m`\n // @return `a * b % m`\n unsigned int mul(unsigned int a, unsigned int b) const {\n // [1] m = 1\n // a = b = im = 0, so okay\n // [2] m >= 2\n // im = ceil(2^64 / m)\n // -> im * m = 2^64 + r (0 <= r < m)\n // let z = a*b = c*m + d (0 <= c, d < m)\n // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im\n // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2\n // ((ab * im) >> 64) == c or c + 1\n unsigned long long z = a;\n z *= b;\n#ifdef _MSC_VER\n unsigned long long x;\n _umul128(z, im, &x);\n#else\n unsigned long long x =\n (unsigned long long)(((unsigned __int128)(z)*im) >> 64);\n#endif\n unsigned int v = (unsigned int)(z - x * _m);\n if (_m <= v) v += _m;\n return v;\n }\n};\n// @param n `0 <= n`\n// @param m `1 <= m`\n// @return `(x ** n) % m`\nconstexpr long long pow_mod_constexpr(long long x, long long n, int m) {\n if (m == 1) return 0;\n unsigned int _m = (unsigned int)(m);\n unsigned long long r = 1;\n unsigned long long y = safe_mod(x, m);\n while (n) {\n if (n & 1) r = (r * y) % _m;\n y = (y * y) % _m;\n n >>= 1;\n }\n return r;\n}\n// Reference:\n// M. Forisek and J. Jancina,\n// Fast Primality Testing for Integers That Fit into a Machine Word\n// @param n `0 <= n`\nconstexpr bool is_prime_constexpr(int n) {\n if (n <= 1) return false;\n if (n == 2 || n == 7 || n == 61) return true;\n if (n % 2 == 0) return false;\n long long d = n - 1;\n while (d % 2 == 0) d /= 2;\n for (long long a : {2, 7, 61}) {\n long long t = d;\n long long y = pow_mod_constexpr(a, t, n);\n while (t != n - 1 && y != 1 && y != n - 1) {\n y = y * y % n;\n t <<= 1;\n }\n if (y != n - 1 && t % 2 == 0) {\n return false;\n }\n }\n return true;\n}\ntemplate constexpr bool is_prime = is_prime_constexpr(n);\n// @param b `1 <= b`\n// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g\nconstexpr std::pair inv_gcd(long long a, long long b) {\n a = safe_mod(a, b);\n if (a == 0) return {b, 0};\n // Contracts:\n // [1] s - m0 * a = 0 (mod b)\n // [2] t - m1 * a = 0 (mod b)\n // [3] s * |m1| + t * |m0| <= b\n long long s = b, t = a;\n long long m0 = 0, m1 = 1;\n while (t) {\n long long u = s / t;\n s -= t * u;\n m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b\n // [3]:\n // (s - t * u) * |m1| + t * |m0 - m1 * u|\n // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)\n // = s * |m1| + t * |m0| <= b\n auto tmp = s;\n s = t;\n t = tmp;\n tmp = m0;\n m0 = m1;\n m1 = tmp;\n }\n // by [3]: |m0| <= b/g\n // by g != b: |m0| < b/g\n if (m0 < 0) m0 += b / s;\n return {s, m0};\n}\n// Compile time primitive root\n// @param m must be prime\n// @return primitive root (and minimum in now)\nconstexpr int primitive_root_constexpr(int m) {\n if (m == 2) return 1;\n if (m == 167772161) return 3;\n if (m == 469762049) return 3;\n if (m == 754974721) return 11;\n if (m == 998244353) return 3;\n int divs[20] = {};\n divs[0] = 2;\n int cnt = 1;\n int x = (m - 1) / 2;\n while (x % 2 == 0) x /= 2;\n for (int i = 3; (long long)(i)*i <= x; i += 2) {\n if (x % i == 0) {\n divs[cnt++] = i;\n while (x % i == 0) {\n x /= i;\n }\n }\n }\n if (x > 1) {\n divs[cnt++] = x;\n }\n for (int g = 2;; g++) {\n bool ok = true;\n for (int i = 0; i < cnt; i++) {\n if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {\n ok = false;\n break;\n }\n }\n if (ok) return g;\n }\n}\ntemplate constexpr int primitive_root = primitive_root_constexpr(m);\n} // namespace internal\n} // namespace atcoder\n#endif // ATCODER_INTERNAL_MATH_HPP\n#ifndef ATCODER_INTERNAL_QUEUE_HPP\n#define ATCODER_INTERNAL_QUEUE_HPP 1\n#include \nnamespace atcoder {\nnamespace internal {\ntemplate struct simple_queue {\n std::vector payload;\n int pos = 0;\n void reserve(int n) { payload.reserve(n); }\n int size() const { return int(payload.size()) - pos; }\n bool empty() const { return pos == int(payload.size()); }\n void push(const T& t) { payload.push_back(t); }\n T& front() { return payload[pos]; }\n void clear() {\n payload.clear();\n pos = 0;\n }\n void pop() { pos++; }\n};\n} // namespace internal\n} // namespace atcoder\n#endif // ATCODER_INTERNAL_QUEUE_HPP\n#ifndef ATCODER_INTERNAL_SCC_HPP\n#define ATCODER_INTERNAL_SCC_HPP 1\n#include \n#include \n#include \nnamespace atcoder {\nnamespace internal {\ntemplate struct csr {\n std::vector start;\n std::vector elist;\n csr(int n, const std::vector>& edges)\n : start(n + 1), elist(edges.size()) {\n for (auto e : edges) {\n start[e.first + 1]++;\n }\n for (int i = 1; i <= n; i++) {\n start[i] += start[i - 1];\n }\n auto counter = start;\n for (auto e : edges) {\n elist[counter[e.first]++] = e.second;\n }\n }\n};\n// Reference:\n// R. Tarjan,\n// Depth-First Search and Linear Graph Algorithms\nstruct scc_graph {\n public:\n scc_graph(int n) : _n(n) {}\n int num_vertices() { return _n; }\n void add_edge(int from, int to) { edges.push_back({from, {to}}); }\n // @return pair of (# of scc, scc id)\n std::pair> scc_ids() {\n auto g = csr(_n, edges);\n int now_ord = 0, group_num = 0;\n std::vector visited, low(_n), ord(_n, -1), ids(_n);\n visited.reserve(_n);\n auto dfs = [&](auto self, int v) -> void {\n low[v] = ord[v] = now_ord++;\n visited.push_back(v);\n for (int i = g.start[v]; i < g.start[v + 1]; i++) {\n auto to = g.elist[i].to;\n if (ord[to] == -1) {\n self(self, to);\n low[v] = std::min(low[v], low[to]);\n } else {\n low[v] = std::min(low[v], ord[to]);\n }\n }\n if (low[v] == ord[v]) {\n while (true) {\n int u = visited.back();\n visited.pop_back();\n ord[u] = _n;\n ids[u] = group_num;\n if (u == v) break;\n }\n group_num++;\n }\n };\n for (int i = 0; i < _n; i++) {\n if (ord[i] == -1) dfs(dfs, i);\n }\n for (auto& x : ids) {\n x = group_num - 1 - x;\n }\n return {group_num, ids};\n }\n std::vector> scc() {\n auto ids = scc_ids();\n int group_num = ids.first;\n std::vector counts(group_num);\n for (auto x : ids.second) counts[x]++;\n std::vector> groups(ids.first);\n for (int i = 0; i < group_num; i++) {\n groups[i].reserve(counts[i]);\n }\n for (int i = 0; i < _n; i++) {\n groups[ids.second[i]].push_back(i);\n }\n return groups;\n }\n private:\n int _n;\n struct edge {\n int to;\n };\n std::vector> edges;\n};\n} // namespace internal\n} // namespace atcoder\n#endif // ATCODER_INTERNAL_SCC_HPP\n#ifndef ATCODER_INTERNAL_TYPE_TRAITS_HPP\n#define ATCODER_INTERNAL_TYPE_TRAITS_HPP 1\n#include \n#include \n#include \nnamespace atcoder {\nnamespace internal {\n#ifndef _MSC_VER\ntemplate \nusing is_signed_int128 =\n typename std::conditional::value ||\n std::is_same::value,\n std::true_type,\n std::false_type>::type;\ntemplate \nusing is_unsigned_int128 =\n typename std::conditional::value ||\n std::is_same::value,\n std::true_type,\n std::false_type>::type;\ntemplate \nusing make_unsigned_int128 =\n typename std::conditional::value,\n __uint128_t,\n unsigned __int128>;\ntemplate \nusing is_integral = typename std::conditional::value ||\n is_signed_int128::value ||\n is_unsigned_int128::value,\n std::true_type,\n std::false_type>::type;\ntemplate \nusing is_signed_int = typename std::conditional<(is_integral::value &&\n std::is_signed::value) ||\n is_signed_int128::value,\n std::true_type,\n std::false_type>::type;\ntemplate \nusing is_unsigned_int =\n typename std::conditional<(is_integral::value &&\n std::is_unsigned::value) ||\n is_unsigned_int128::value,\n std::true_type,\n std::false_type>::type;\ntemplate \nusing to_unsigned = typename std::conditional<\n is_signed_int128::value,\n make_unsigned_int128,\n typename std::conditional::value,\n std::make_unsigned,\n std::common_type>::type>::type;\n#else\ntemplate using is_integral = typename std::is_integral;\ntemplate \nusing is_signed_int =\n typename std::conditional::value && std::is_signed::value,\n std::true_type,\n std::false_type>::type;\ntemplate \nusing is_unsigned_int =\n typename std::conditional::value &&\n std::is_unsigned::value,\n std::true_type,\n std::false_type>::type;\ntemplate \nusing to_unsigned = typename std::conditional::value,\n std::make_unsigned,\n std::common_type>::type;\n#endif\ntemplate \nusing is_signed_int_t = std::enable_if_t::value>;\ntemplate \nusing is_unsigned_int_t = std::enable_if_t::value>;\ntemplate using to_unsigned_t = typename to_unsigned::type;\n} // namespace internal\n} // namespace atcoder\n#endif // ATCODER_INTERNAL_TYPE_TRAITS_HPP\n#ifndef ATCODER_MODINT_HPP\n#define ATCODER_MODINT_HPP 1\n#include \n#include \n#include \n#ifdef _MSC_VER\n#include \n#endif\nnamespace atcoder {\nnamespace internal {\nstruct modint_base {};\nstruct static_modint_base : modint_base {};\ntemplate using is_modint = std::is_base_of;\ntemplate using is_modint_t = std::enable_if_t::value>;\n} // namespace internal\ntemplate * = nullptr>\nstruct static_modint : internal::static_modint_base {\n using mint = static_modint;\n public:\n static constexpr int mod() { return m; }\n static mint raw(int v) {\n mint x;\n x._v = v;\n return x;\n }\n static_modint() : _v(0) {}\n template * = nullptr>\n static_modint(T v) {\n long long x = (long long)(v % (long long)(umod()));\n if (x < 0) x += umod();\n _v = (unsigned int)(x);\n }\n template * = nullptr>\n static_modint(T v) {\n _v = (unsigned int)(v % umod());\n }\n static_modint(bool v) { _v = ((unsigned int)(v) % umod()); }\n unsigned int val() const { return _v; }\n mint& operator++() {\n _v++;\n if (_v == umod()) _v = 0;\n return *this;\n }\n mint& operator--() {\n if (_v == 0) _v = umod();\n _v--;\n return *this;\n }\n mint operator++(int) {\n mint result = *this;\n ++*this;\n return result;\n }\n mint operator--(int) {\n mint result = *this;\n --*this;\n return result;\n }\n mint& operator+=(const mint& rhs) {\n _v += rhs._v;\n if (_v >= umod()) _v -= umod();\n return *this;\n }\n mint& operator-=(const mint& rhs) {\n _v -= rhs._v;\n if (_v >= umod()) _v += umod();\n return *this;\n }\n mint& operator*=(const mint& rhs) {\n unsigned long long z = _v;\n z *= rhs._v;\n _v = (unsigned int)(z % umod());\n return *this;\n }\n mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }\n mint operator+() const { return *this; }\n mint operator-() const { return mint() - *this; }\n mint pow(long long n) const {\n assert(0 <= n);\n mint x = *this, r = 1;\n while (n) {\n if (n & 1) r *= x;\n x *= x;\n n >>= 1;\n }\n return r;\n }\n mint inv() const {\n if (prime) {\n assert(_v);\n return pow(umod() - 2);\n } else {\n auto eg = internal::inv_gcd(_v, m);\n assert(eg.first == 1);\n return eg.second;\n }\n }\n friend mint operator+(const mint& lhs, const mint& rhs) {\n return mint(lhs) += rhs;\n }\n friend mint operator-(const mint& lhs, const mint& rhs) {\n return mint(lhs) -= rhs;\n }\n friend mint operator*(const mint& lhs, const mint& rhs) {\n return mint(lhs) *= rhs;\n }\n friend mint operator/(const mint& lhs, const mint& rhs) {\n return mint(lhs) /= rhs;\n }\n friend bool operator==(const mint& lhs, const mint& rhs) {\n return lhs._v == rhs._v;\n }\n friend bool operator!=(const mint& lhs, const mint& rhs) {\n return lhs._v != rhs._v;\n }\n private:\n unsigned int _v;\n static constexpr unsigned int umod() { return m; }\n static constexpr bool prime = internal::is_prime;\n};\ntemplate struct dynamic_modint : internal::modint_base {\n using mint = dynamic_modint;\n public:\n static int mod() { return (int)(bt.umod()); }\n static void set_mod(int m) {\n assert(1 <= m);\n bt = internal::barrett(m);\n }\n static mint raw(int v) {\n mint x;\n x._v = v;\n return x;\n }\n dynamic_modint() : _v(0) {}\n template * = nullptr>\n dynamic_modint(T v) {\n long long x = (long long)(v % (long long)(mod()));\n if (x < 0) x += mod();\n _v = (unsigned int)(x);\n }\n template * = nullptr>\n dynamic_modint(T v) {\n _v = (unsigned int)(v % mod());\n }\n dynamic_modint(bool v) { _v = ((unsigned int)(v) % mod()); }\n unsigned int val() const { return _v; }\n mint& operator++() {\n _v++;\n if (_v == umod()) _v = 0;\n return *this;\n }\n mint& operator--() {\n if (_v == 0) _v = umod();\n _v--;\n return *this;\n }\n mint operator++(int) {\n mint result = *this;\n ++*this;\n return result;\n }\n mint operator--(int) {\n mint result = *this;\n --*this;\n return result;\n }\n mint& operator+=(const mint& rhs) {\n _v += rhs._v;\n if (_v >= umod()) _v -= umod();\n return *this;\n }\n mint& operator-=(const mint& rhs) {\n _v += mod() - rhs._v;\n if (_v >= umod()) _v -= umod();\n return *this;\n }\n mint& operator*=(const mint& rhs) {\n _v = bt.mul(_v, rhs._v);\n return *this;\n }\n mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }\n mint operator+() const { return *this; }\n mint operator-() const { return mint() - *this; }\n mint pow(long long n) const {\n assert(0 <= n);\n mint x = *this, r = 1;\n while (n) {\n if (n & 1) r *= x;\n x *= x;\n n >>= 1;\n }\n return r;\n }\n mint inv() const {\n auto eg = internal::inv_gcd(_v, mod());\n assert(eg.first == 1);\n return eg.second;\n }\n friend mint operator+(const mint& lhs, const mint& rhs) {\n return mint(lhs) += rhs;\n }\n friend mint operator-(const mint& lhs, const mint& rhs) {\n return mint(lhs) -= rhs;\n }\n friend mint operator*(const mint& lhs, const mint& rhs) {\n return mint(lhs) *= rhs;\n }\n friend mint operator/(const mint& lhs, const mint& rhs) {\n return mint(lhs) /= rhs;\n }\n friend bool operator==(const mint& lhs, const mint& rhs) {\n return lhs._v == rhs._v;\n }\n friend bool operator!=(const mint& lhs, const mint& rhs) {\n return lhs._v != rhs._v;\n }\n private:\n unsigned int _v;\n static internal::barrett bt;\n static unsigned int umod() { return bt.umod(); }\n};\ntemplate internal::barrett dynamic_modint::bt = 998244353;\nusing modint998244353 = static_modint<998244353>;\nusing modint1000000007 = static_modint<1000000007>;\nusing modint = dynamic_modint<-1>;\nnamespace internal {\ntemplate \nusing is_static_modint = std::is_base_of;\ntemplate \nusing is_static_modint_t = std::enable_if_t::value>;\ntemplate struct is_dynamic_modint : public std::false_type {};\ntemplate \nstruct is_dynamic_modint> : public std::true_type {};\ntemplate \nusing is_dynamic_modint_t = std::enable_if_t::value>;\n} // namespace internal\n} // namespace atcoder\n#endif // ATCODER_MODINT_HPP\n#ifndef ATCODER_CONVOLUTION_HPP\n#define ATCODER_CONVOLUTION_HPP 1\n#include \n#include \n#include \n#include \n#include \nnamespace atcoder {\nnamespace internal {\ntemplate * = nullptr>\nvoid butterfly(std::vector& a) {\n static constexpr int g = internal::primitive_root;\n int n = int(a.size());\n int h = internal::ceil_pow2(n);\n static bool first = true;\n static mint sum_e[30]; // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]\n if (first) {\n first = false;\n mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1\n int cnt2 = bsf(mint::mod() - 1);\n mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();\n for (int i = cnt2; i >= 2; i--) {\n // e^(2^i) == 1\n es[i - 2] = e;\n ies[i - 2] = ie;\n e *= e;\n ie *= ie;\n }\n mint now = 1;\n for (int i = 0; i < cnt2 - 2; i++) {\n sum_e[i] = es[i] * now;\n now *= ies[i];\n }\n }\n for (int ph = 1; ph <= h; ph++) {\n int w = 1 << (ph - 1), p = 1 << (h - ph);\n mint now = 1;\n for (int s = 0; s < w; s++) {\n int offset = s << (h - ph + 1);\n for (int i = 0; i < p; i++) {\n auto l = a[i + offset];\n auto r = a[i + offset + p] * now;\n a[i + offset] = l + r;\n a[i + offset + p] = l - r;\n }\n now *= sum_e[bsf(~(unsigned int)(s))];\n }\n }\n}\ntemplate * = nullptr>\nvoid butterfly_inv(std::vector& a) {\n static constexpr int g = internal::primitive_root;\n int n = int(a.size());\n int h = internal::ceil_pow2(n);\n static bool first = true;\n static mint sum_ie[30]; // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]\n if (first) {\n first = false;\n mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1\n int cnt2 = bsf(mint::mod() - 1);\n mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();\n for (int i = cnt2; i >= 2; i--) {\n // e^(2^i) == 1\n es[i - 2] = e;\n ies[i - 2] = ie;\n e *= e;\n ie *= ie;\n }\n mint now = 1;\n for (int i = 0; i < cnt2 - 2; i++) {\n sum_ie[i] = ies[i] * now;\n now *= es[i];\n }\n }\n for (int ph = h; ph >= 1; ph--) {\n int w = 1 << (ph - 1), p = 1 << (h - ph);\n mint inow = 1;\n for (int s = 0; s < w; s++) {\n int offset = s << (h - ph + 1);\n for (int i = 0; i < p; i++) {\n auto l = a[i + offset];\n auto r = a[i + offset + p];\n a[i + offset] = l + r;\n a[i + offset + p] =\n (unsigned long long)(mint::mod() + l.val() - r.val()) *\n inow.val();\n }\n inow *= sum_ie[bsf(~(unsigned int)(s))];\n }\n }\n}\n} // namespace internal\ntemplate * = nullptr>\nstd::vector convolution(std::vector a, std::vector b) {\n int n = int(a.size()), m = int(b.size());\n if (!n || !m) return {};\n if (std::min(n, m) <= 60) {\n if (n < m) {\n std::swap(n, m);\n std::swap(a, b);\n }\n std::vector ans(n + m - 1);\n for (int i = 0; i < n; i++) {\n for (int j = 0; j < m; j++) {\n ans[i + j] += a[i] * b[j];\n }\n }\n return ans;\n }\n int z = 1 << internal::ceil_pow2(n + m - 1);\n a.resize(z);\n internal::butterfly(a);\n b.resize(z);\n internal::butterfly(b);\n for (int i = 0; i < z; i++) {\n a[i] *= b[i];\n }\n internal::butterfly_inv(a);\n a.resize(n + m - 1);\n mint iz = mint(z).inv();\n for (int i = 0; i < n + m - 1; i++) a[i] *= iz;\n return a;\n}\ntemplate ::value>* = nullptr>\nstd::vector convolution(const std::vector& a, const std::vector& b) {\n int n = int(a.size()), m = int(b.size());\n if (!n || !m) return {};\n using mint = static_modint;\n std::vector a2(n), b2(m);\n for (int i = 0; i < n; i++) {\n a2[i] = mint(a[i]);\n }\n for (int i = 0; i < m; i++) {\n b2[i] = mint(b[i]);\n }\n auto c2 = convolution(move(a2), move(b2));\n std::vector c(n + m - 1);\n for (int i = 0; i < n + m - 1; i++) {\n c[i] = c2[i].val();\n }\n return c;\n}\nstd::vector convolution_ll(const std::vector& a,\n const std::vector& b) {\n int n = int(a.size()), m = int(b.size());\n if (!n || !m) return {};\n static constexpr unsigned long long MOD1 = 754974721; // 2^24\n static constexpr unsigned long long MOD2 = 167772161; // 2^25\n static constexpr unsigned long long MOD3 = 469762049; // 2^26\n static constexpr unsigned long long M2M3 = MOD2 * MOD3;\n static constexpr unsigned long long M1M3 = MOD1 * MOD3;\n static constexpr unsigned long long M1M2 = MOD1 * MOD2;\n static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;\n static constexpr unsigned long long i1 =\n internal::inv_gcd(MOD2 * MOD3, MOD1).second;\n static constexpr unsigned long long i2 =\n internal::inv_gcd(MOD1 * MOD3, MOD2).second;\n static constexpr unsigned long long i3 =\n internal::inv_gcd(MOD1 * MOD2, MOD3).second;\n auto c1 = convolution(a, b);\n auto c2 = convolution(a, b);\n auto c3 = convolution(a, b);\n std::vector c(n + m - 1);\n for (int i = 0; i < n + m - 1; i++) {\n unsigned long long x = 0;\n x += (c1[i] * i1) % MOD1 * M2M3;\n x += (c2[i] * i2) % MOD2 * M1M3;\n x += (c3[i] * i3) % MOD3 * M1M2;\n // B = 2^63, -B <= x, r(real value) < B\n // (x, x - M, x - 2M, or x - 3M) = r (mod 2B)\n // r = c1[i] (mod MOD1)\n // focus on MOD1\n // r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)\n // r = x,\n // x - M' + (0 or 2B),\n // x - 2M' + (0, 2B or 4B),\n // x - 3M' + (0, 2B, 4B or 6B) (without mod!)\n // (r - x) = 0, (0)\n // - M' + (0 or 2B), (1)\n // -2M' + (0 or 2B or 4B), (2)\n // -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)\n // we checked that\n // ((1) mod MOD1) mod 5 = 2\n // ((2) mod MOD1) mod 5 = 3\n // ((3) mod MOD1) mod 5 = 4\n long long diff =\n c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));\n if (diff < 0) diff += MOD1;\n static constexpr unsigned long long offset[5] = {\n 0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};\n x -= offset[diff % 5];\n c[i] = x;\n }\n return c;\n}\n} // namespace atcoder\n#endif // ATCODER_CONVOLUTION_HPP\n#ifndef ATCODER_DSU_HPP\n#define ATCODER_DSU_HPP 1\n#include \n#include \n#include \nnamespace atcoder {\n// Implement (union by size) + (path compression)\n// Reference:\n// Zvi Galil and Giuseppe F. Italiano,\n// Data structures and algorithms for disjoint set union problems\nstruct dsu {\n public:\n dsu() : _n(0) {}\n dsu(int n) : _n(n), parent_or_size(n, -1) {}\n int merge(int a, int b) {\n assert(0 <= a && a < _n); \n assert(0 <= b && b < _n);\n int x = leader(a), y = leader(b);\n if (x == y) return x;\n if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y);\n parent_or_size[x] += parent_or_size[y];\n parent_or_size[y] = x;\n return x;\n }\n bool same(int a, int b) {\n assert(0 <= a && a < _n);\n assert(0 <= b && b < _n);\n return leader(a) == leader(b);\n }\n int leader(int a) {\n assert(0 <= a && a < _n);\n if (parent_or_size[a] < 0) return a;\n return parent_or_size[a] = leader(parent_or_size[a]);\n }\n int size(int a) {\n assert(0 <= a && a < _n);\n return -parent_or_size[leader(a)];\n }\n std::vector> groups() {\n std::vector leader_buf(_n), group_size(_n);\n for (int i = 0; i < _n; i++) {\n leader_buf[i] = leader(i);\n group_size[leader_buf[i]]++;\n }\n std::vector> result(_n);\n for (int i = 0; i < _n; i++) {\n result[i].reserve(group_size[i]);\n }\n for (int i = 0; i < _n; i++) {\n result[leader_buf[i]].push_back(i);\n }\n result.erase(\n std::remove_if(result.begin(), result.end(),\n [&](const std::vector& v) { return v.empty(); }),\n result.end());\n return result;\n }\n private:\n int _n;\n // root node: -1 * component size\n // otherwise: parent\n std::vector parent_or_size;\n};\n} // namespace atcoder\n#endif // ATCODER_DSU_HPP\n#ifndef ATCODER_FENWICKTREE_HPP\n#define ATCODER_FENWICKTREE_HPP 1\n#include \n#include \nnamespace atcoder {\n// Reference: https://en.wikipedia.org/wiki/Fenwick_tree\ntemplate struct fenwick_tree {\n using U = internal::to_unsigned_t;\n public:\n fenwick_tree() : _n(0) {}\n fenwick_tree(int n) : _n(n), data(n) {}\n void add(int p, T x) {\n assert(0 <= p && p < _n);\n p++;\n while (p <= _n) {\n data[p - 1] += U(x);\n p += p & -p;\n }\n }\n T sum(int l, int r) {\n assert(0 <= l && l <= r && r <= _n);\n return sum(r) - sum(l);\n }\n private:\n int _n;\n std::vector data;\n U sum(int r) {\n U s = 0;\n while (r > 0) {\n s += data[r - 1];\n r -= r & -r;\n }\n return s;\n }\n};\n} // namespace atcoder\n#endif // ATCODER_FENWICKTREE_HPP\n#ifndef ATCODER_LAZYSEGTREE_HPP\n#define ATCODER_LAZYSEGTREE_HPP 1\n#include \n#include \n#include \n#include \nnamespace atcoder {\ntemplate \nstruct lazy_segtree {\n public:\n lazy_segtree() : lazy_segtree(0) {}\n lazy_segtree(int n) : lazy_segtree(std::vector(n, e())) {}\n lazy_segtree(const std::vector& v) : _n(int(v.size())) {\n log = internal::ceil_pow2(_n);\n size = 1 << log;\n d = std::vector(2 * size, e());\n lz = std::vector(size, id());\n for (int i = 0; i < _n; i++) d[size + i] = v[i];\n for (int i = size - 1; i >= 1; i--) {\n update(i);\n }\n }\n void set(int p, S x) {\n assert(0 <= p && p < _n);\n p += size;\n for (int i = log; i >= 1; i--) push(p >> i);\n d[p] = x;\n for (int i = 1; i <= log; i++) update(p >> i);\n }\n S get(int p) {\n assert(0 <= p && p < _n);\n p += size;\n for (int i = log; i >= 1; i--) push(p >> i);\n return d[p];\n }\n S prod(int l, int r) {\n assert(0 <= l && l <= r && r <= _n);\n if (l == r) return e();\n l += size;\n r += size;\n for (int i = log; i >= 1; i--) {\n if (((l >> i) << i) != l) push(l >> i);\n if (((r >> i) << i) != r) push(r >> i);\n }\n S sml = e(), smr = e();\n while (l < r) {\n if (l & 1) sml = op(sml, d[l++]);\n if (r & 1) smr = op(d[--r], smr);\n l >>= 1;\n r >>= 1;\n }\n return op(sml, smr);\n }\n S all_prod() { return d[1]; }\n void apply(int p, F f) {\n assert(0 <= p && p < _n);\n p += size;\n for (int i = log; i >= 1; i--) push(p >> i);\n d[p] = mapping(f, d[p]);\n for (int i = 1; i <= log; i++) update(p >> i);\n }\n void apply(int l, int r, F f) {\n assert(0 <= l && l <= r && r <= _n);\n if (l == r) return;\n l += size;\n r += size;\n for (int i = log; i >= 1; i--) {\n if (((l >> i) << i) != l) push(l >> i);\n if (((r >> i) << i) != r) push((r - 1) >> i);\n }\n {\n int l2 = l, r2 = r;\n while (l < r) {\n if (l & 1) all_apply(l++, f);\n if (r & 1) all_apply(--r, f);\n l >>= 1;\n r >>= 1;\n }\n l = l2;\n r = r2;\n }\n for (int i = 1; i <= log; i++) {\n if (((l >> i) << i) != l) update(l >> i);\n if (((r >> i) << i) != r) update((r - 1) >> i);\n }\n }\n template int max_right(int l) {\n return max_right(l, [](S x) { return g(x); });\n }\n template int max_right(int l, G g) {\n assert(0 <= l && l <= _n);\n assert(g(e()));\n if (l == _n) return _n;\n l += size;\n for (int i = log; i >= 1; i--) push(l >> i);\n S sm = e();\n do {\n while (l % 2 == 0) l >>= 1;\n if (!g(op(sm, d[l]))) {\n while (l < size) {\n push(l);\n l = (2 * l);\n if (g(op(sm, d[l]))) {\n sm = op(sm, d[l]);\n l++;\n }\n }\n return l - size;\n }\n sm = op(sm, d[l]);\n l++;\n } while ((l & -l) != l);\n return _n;\n }\n template int min_left(int r) {\n return min_left(r, [](S x) { return g(x); });\n }\n template int min_left(int r, G g) {\n assert(0 <= r && r <= _n);\n assert(g(e()));\n if (r == 0) return 0;\n r += size;\n for (int i = log; i >= 1; i--) push((r - 1) >> i);\n S sm = e();\n do {\n r--;\n while (r > 1 && (r % 2)) r >>= 1;\n if (!g(op(d[r], sm))) {\n while (r < size) {\n push(r);\n r = (2 * r + 1);\n if (g(op(d[r], sm))) {\n sm = op(d[r], sm);\n r--;\n }\n }\n return r + 1 - size;\n }\n sm = op(d[r], sm);\n } while ((r & -r) != r);\n return 0;\n }\n private:\n int _n, size, log;\n std::vector d;\n std::vector lz;\n void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }\n void all_apply(int k, F f) {\n d[k] = mapping(f, d[k]);\n if (k < size) lz[k] = composition(f, lz[k]);\n }\n void push(int k) {\n all_apply(2 * k, lz[k]);\n all_apply(2 * k + 1, lz[k]);\n lz[k] = id();\n }\n};\n} // namespace atcoder\n#endif // ATCODER_LAZYSEGTREE_HPP\n#ifndef ATCODER_MATH_HPP\n#define ATCODER_MATH_HPP 1\n#include \n#include \n#include \n#include \nnamespace atcoder {\nlong long pow_mod(long long x, long long n, int m) {\n assert(0 <= n && 1 <= m);\n if (m == 1) return 0;\n internal::barrett bt((unsigned int)(m));\n unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m));\n while (n) {\n if (n & 1) r = bt.mul(r, y);\n y = bt.mul(y, y);\n n >>= 1;\n }\n return r;\n}\nlong long inv_mod(long long x, long long m) {\n assert(1 <= m);\n auto z = internal::inv_gcd(x, m);\n assert(z.first == 1);\n return z.second;\n}\n// (rem, mod)\nstd::pair crt(const std::vector& r,\n const std::vector& m) {\n assert(r.size() == m.size());\n int n = int(r.size());\n // Contracts: 0 <= r0 < m0\n long long r0 = 0, m0 = 1;\n for (int i = 0; i < n; i++) {\n assert(1 <= m[i]);\n long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i];\n if (m0 < m1) {\n std::swap(r0, r1);\n std::swap(m0, m1);\n }\n if (m0 % m1 == 0) {\n if (r0 % m1 != r1) return {0, 0};\n continue;\n }\n // assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1)\n // (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1));\n // r2 % m0 = r0\n // r2 % m1 = r1\n // -> (r0 + x*m0) % m1 = r1\n // -> x*u0*g % (u1*g) = (r1 - r0) (u0*g = m0, u1*g = m1)\n // -> x = (r1 - r0) / g * inv(u0) (mod u1)\n // im = inv(u0) (mod u1) (0 <= im < u1)\n long long g, im;\n std::tie(g, im) = internal::inv_gcd(m0, m1);\n long long u1 = (m1 / g);\n // |r1 - r0| < (m0 + m1) <= lcm(m0, m1)\n if ((r1 - r0) % g) return {0, 0};\n // u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1)\n long long x = (r1 - r0) / g % u1 * im % u1;\n // |r0| + |m0 * x|\n // < m0 + m0 * (u1 - 1)\n // = m0 + m0 * m1 / g - m0\n // = lcm(m0, m1)\n r0 += x * m0;\n m0 *= u1; // -> lcm(m0, m1)\n if (r0 < 0) r0 += m0;\n }\n return {r0, m0};\n}\nlong long floor_sum(long long n, long long m, long long a, long long b) {\n long long ans = 0; \n if (a >= m) {\n ans += (n - 1) * n * (a / m) / 2;\n a %= m;\n }\n if (b >= m) {\n ans += n * (b / m);\n b %= m;\n }\n long long y_max = (a * n + b) / m, x_max = (y_max * m - b);\n if (y_max == 0) return ans;\n ans += (n - (x_max + a - 1) / a) * y_max;\n ans += floor_sum(y_max, a, m, (a - x_max % a) % a);\n return ans;\n}\n} // namespace atcoder\n#endif // ATCODER_MATH_HPP\n#ifndef ATCODER_MAXFLOW_HPP\n#define ATCODER_MAXFLOW_HPP 1\n#include \n#include \n#include \n#include \n#include \nnamespace atcoder {\ntemplate struct mf_graph {\n public:\n mf_graph() : _n(0) {}\n mf_graph(int n) : _n(n), g(n) {}\n int add_edge(int from, int to, Cap cap) {\n assert(0 <= from && from < _n);\n assert(0 <= to && to < _n);\n assert(0 <= cap);\n int m = int(pos.size());\n pos.push_back({from, int(g[from].size())});\n g[from].push_back(_edge{to, int(g[to].size()), cap});\n g[to].push_back(_edge{from, int(g[from].size()) - 1, 0});\n return m;\n }\n struct edge {\n int from, to;\n Cap cap, flow;\n };\n edge get_edge(int i) {\n int m = int(pos.size());\n assert(0 <= i && i < m);\n auto _e = g[pos[i].first][pos[i].second];\n auto _re = g[_e.to][_e.rev];\n return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};\n }\n std::vector edges() {\n int m = int(pos.size());\n std::vector result;\n for (int i = 0; i < m; i++) {\n result.push_back(get_edge(i));\n }\n return result;\n }\n void change_edge(int i, Cap new_cap, Cap new_flow) {\n int m = int(pos.size());\n assert(0 <= i && i < m);\n assert(0 <= new_flow && new_flow <= new_cap);\n auto& _e = g[pos[i].first][pos[i].second];\n auto& _re = g[_e.to][_e.rev];\n _e.cap = new_cap - new_flow;\n _re.cap = new_flow;\n }\n Cap flow(int s, int t) {\n return flow(s, t, std::numeric_limits::max());\n }\n Cap flow(int s, int t, Cap flow_limit) {\n assert(0 <= s && s < _n);\n assert(0 <= t && t < _n);\n std::vector level(_n), iter(_n);\n internal::simple_queue que;\n auto bfs = [&]() {\n std::fill(level.begin(), level.end(), -1);\n level[s] = 0;\n que.clear();\n que.push(s);\n while (!que.empty()) {\n int v = que.front();\n que.pop();\n for (auto e : g[v]) {\n if (e.cap == 0 || level[e.to] >= 0) continue;\n level[e.to] = level[v] + 1;\n if (e.to == t) return;\n que.push(e.to);\n }\n }\n };\n auto dfs = [&](auto self, int v, Cap up) {\n if (v == s) return up;\n Cap res = 0;\n int level_v = level[v];\n for (int& i = iter[v]; i < int(g[v].size()); i++) {\n _edge& e = g[v][i];\n if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;\n Cap d =\n self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));\n if (d <= 0) continue;\n g[v][i].cap += d;\n g[e.to][e.rev].cap -= d;\n res += d;\n if (res == up) break;\n }\n return res;\n };\n Cap flow = 0;\n while (flow < flow_limit) {\n bfs();\n if (level[t] == -1) break;\n std::fill(iter.begin(), iter.end(), 0);\n while (flow < flow_limit) {\n Cap f = dfs(dfs, t, flow_limit - flow);\n if (!f) break;\n flow += f;\n }\n }\n return flow;\n }\n std::vector min_cut(int s) {\n std::vector visited(_n);\n internal::simple_queue que;\n que.push(s);\n while (!que.empty()) {\n int p = que.front();\n que.pop();\n visited[p] = true;\n for (auto e : g[p]) {\n if (e.cap && !visited[e.to]) {\n visited[e.to] = true;\n que.push(e.to);\n }\n }\n }\n return visited;\n }\n private:\n int _n;\n struct _edge {\n int to, rev;\n Cap cap;\n };\n std::vector> pos;\n std::vector> g;\n};\n} // namespace atcoder\n#endif // ATCODER_MAXFLOW_HPP\n#ifndef ATCODER_MINCOSTFLOW_HPP\n#define ATCODER_MINCOSTFLOW_HPP 1\n#include \n#include \n#include \n#include \n#include \nnamespace atcoder {\ntemplate struct mcf_graph {\n public:\n mcf_graph() {}\n mcf_graph(int n) : _n(n), g(n) {}\n int add_edge(int from, int to, Cap cap, Cost cost) {\n assert(0 <= from && from < _n);\n assert(0 <= to && to < _n);\n int m = int(pos.size());\n pos.push_back({from, int(g[from].size())});\n g[from].push_back(_edge{to, int(g[to].size()), cap, cost});\n g[to].push_back(_edge{from, int(g[from].size()) - 1, 0, -cost});\n return m;\n }\n struct edge {\n int from, to;\n Cap cap, flow;\n Cost cost;\n };\n edge get_edge(int i) {\n int m = int(pos.size());\n assert(0 <= i && i < m);\n auto _e = g[pos[i].first][pos[i].second];\n auto _re = g[_e.to][_e.rev];\n return edge{\n pos[i].first, _e.to, _e.cap + _re.cap, _re.cap, _e.cost,\n };\n }\n std::vector edges() {\n int m = int(pos.size());\n std::vector result(m);\n for (int i = 0; i < m; i++) {\n result[i] = get_edge(i);\n }\n return result;\n }\n std::pair flow(int s, int t) {\n return flow(s, t, std::numeric_limits::max());\n }\n std::pair flow(int s, int t, Cap flow_limit) {\n return slope(s, t, flow_limit).back();\n }\n std::vector> slope(int s, int t) {\n return slope(s, t, std::numeric_limits::max());\n }\n std::vector> slope(int s, int t, Cap flow_limit) {\n assert(0 <= s && s < _n);\n assert(0 <= t && t < _n);\n assert(s != t);\n // variants (C = maxcost):\n // -(n-1)C <= dual[s] <= dual[i] <= dual[t] = 0\n // reduced cost (= e.cost + dual[e.from] - dual[e.to]) >= 0 for all edge\n std::vector dual(_n, 0), dist(_n);\n std::vector pv(_n), pe(_n);\n std::vector vis(_n);\n auto dual_ref = [&]() {\n std::fill(dist.begin(), dist.end(),\n std::numeric_limits::max());\n std::fill(pv.begin(), pv.end(), -1);\n std::fill(pe.begin(), pe.end(), -1);\n std::fill(vis.begin(), vis.end(), false);\n struct Q {\n Cost key;\n int to;\n bool operator<(Q r) const { return key > r.key; }\n };\n std::priority_queue que;\n dist[s] = 0;\n que.push(Q{0, s});\n while (!que.empty()) {\n int v = que.top().to;\n que.pop();\n if (vis[v]) continue;\n vis[v] = true;\n if (v == t) break;\n // dist[v] = shortest(s, v) + dual[s] - dual[v]\n // dist[v] >= 0 (all reduced cost are positive)\n // dist[v] <= (n-1)C\n for (int i = 0; i < int(g[v].size()); i++) {\n auto e = g[v][i];\n if (vis[e.to] || !e.cap) continue;\n // |-dual[e.to] + dual[v]| <= (n-1)C\n // cost <= C - -(n-1)C + 0 = nC\n Cost cost = e.cost - dual[e.to] + dual[v];\n if (dist[e.to] - dist[v] > cost) {\n dist[e.to] = dist[v] + cost;\n pv[e.to] = v;\n pe[e.to] = i;\n que.push(Q{dist[e.to], e.to});\n }\n }\n }\n if (!vis[t]) {\n return false;\n }\n for (int v = 0; v < _n; v++) {\n if (!vis[v]) continue;\n // dual[v] = dual[v] - dist[t] + dist[v]\n // = dual[v] - (shortest(s, t) + dual[s] - dual[t]) + (shortest(s, v) + dual[s] - dual[v])\n // = - shortest(s, t) + dual[t] + shortest(s, v)\n // = shortest(s, v) - shortest(s, t) >= 0 - (n-1)C\n dual[v] -= dist[t] - dist[v];\n }\n return true;\n };\n Cap flow = 0;\n Cost cost = 0, prev_cost = -1;\n std::vector> result;\n result.push_back({flow, cost});\n while (flow < flow_limit) {\n if (!dual_ref()) break;\n Cap c = flow_limit - flow;\n for (int v = t; v != s; v = pv[v]) {\n c = std::min(c, g[pv[v]][pe[v]].cap);\n }\n for (int v = t; v != s; v = pv[v]) {\n auto& e = g[pv[v]][pe[v]];\n e.cap -= c;\n g[v][e.rev].cap += c;\n }\n Cost d = -dual[s];\n flow += c;\n cost += c * d;\n if (prev_cost == d) {\n result.pop_back();\n }\n result.push_back({flow, cost});\n prev_cost = cost;\n }\n return result;\n }\n private:\n int _n;\n struct _edge {\n int to, rev;\n Cap cap;\n Cost cost;\n };\n std::vector> pos;\n std::vector> g;\n};\n} // namespace atcoder\n#endif // ATCODER_MINCOSTFLOW_HPP\n#ifndef ATCODER_SCC_HPP\n#define ATCODER_SCC_HPP 1\n#include \n#include \n#include \nnamespace atcoder {\nstruct scc_graph {\n public:\n scc_graph() : internal(0) {}\n scc_graph(int n) : internal(n) {}\n void add_edge(int from, int to) {\n int n = internal.num_vertices();\n assert(0 <= from && from < n);\n assert(0 <= to && to < n);\n internal.add_edge(from, to);\n }\n std::vector> scc() { return internal.scc(); }\n private:\n internal::scc_graph internal;\n};\n} // namespace atcoder\n#endif // ATCODER_SCC_HPP\n#ifndef ATCODER_SEGTREE_HPP\n#define ATCODER_SEGTREE_HPP 1\n#include \n#include \n#include \nnamespace atcoder {\ntemplate struct segtree {\n public:\n segtree() : segtree(0) {}\n segtree(int n) : segtree(std::vector(n, e())) {}\n segtree(const std::vector& v) : _n(int(v.size())) {\n log = internal::ceil_pow2(_n);\n size = 1 << log;\n d = std::vector(2 * size, e());\n for (int i = 0; i < _n; i++) d[size + i] = v[i];\n for (int i = size - 1; i >= 1; i--) {\n update(i);\n }\n }\n void set(int p, S x) {\n assert(0 <= p && p < _n);\n p += size;\n d[p] = x;\n for (int i = 1; i <= log; i++) update(p >> i);\n }\n S get(int p) {\n assert(0 <= p && p < _n);\n return d[p + size];\n }\n S prod(int l, int r) {\n assert(0 <= l && l <= r && r <= _n);\n S sml = e(), smr = e();\n l += size;\n r += size;\n while (l < r) {\n if (l & 1) sml = op(sml, d[l++]);\n if (r & 1) smr = op(d[--r], smr);\n l >>= 1;\n r >>= 1;\n }\n return op(sml, smr);\n }\n S all_prod() { return d[1]; }\n template int max_right(int l) {\n return max_right(l, [](S x) { return f(x); });\n }\n template int max_right(int l, F f) {\n assert(0 <= l && l <= _n);\n assert(f(e()));\n if (l == _n) return _n;\n l += size;\n S sm = e();\n do {\n while (l % 2 == 0) l >>= 1;\n if (!f(op(sm, d[l]))) {\n while (l < size) {\n l = (2 * l);\n if (f(op(sm, d[l]))) {\n sm = op(sm, d[l]);\n l++;\n }\n }\n return l - size;\n }\n sm = op(sm, d[l]);\n l++;\n } while ((l & -l) != l);\n return _n;\n }\n template int min_left(int r) {\n return min_left(r, [](S x) { return f(x); });\n }\n template int min_left(int r, F f) {\n assert(0 <= r && r <= _n);\n assert(f(e()));\n if (r == 0) return 0;\n r += size;\n S sm = e();\n do {\n r--;\n while (r > 1 && (r % 2)) r >>= 1;\n if (!f(op(d[r], sm))) {\n while (r < size) {\n r = (2 * r + 1);\n if (f(op(d[r], sm))) {\n sm = op(d[r], sm);\n r--;\n }\n }\n return r + 1 - size;\n }\n sm = op(d[r], sm);\n } while ((r & -r) != r);\n return 0;\n }\n private:\n int _n, size, log;\n std::vector d;\n void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }\n};\n} // namespace atcoder\n#endif // ATCODER_SEGTREE_HPP\n#ifndef ATCODER_STRING_HPP\n#define ATCODER_STRING_HPP 1\n#include \n#include \n#include \n#include \n#include \nnamespace atcoder {\nnamespace internal {\nstd::vector sa_naive(const std::vector& s) {\n int n = int(s.size());\n std::vector sa(n);\n std::iota(sa.begin(), sa.end(), 0);\n std::sort(sa.begin(), sa.end(), [&](int l, int r) {\n if (l == r) return false;\n while (l < n && r < n) {\n if (s[l] != s[r]) return s[l] < s[r];\n l++;\n r++;\n }\n return l == n;\n });\n return sa;\n}\nstd::vector sa_doubling(const std::vector& s) {\n int n = int(s.size());\n std::vector sa(n), rnk = s, tmp(n);\n std::iota(sa.begin(), sa.end(), 0);\n for (int k = 1; k < n; k *= 2) {\n auto cmp = [&](int x, int y) {\n if (rnk[x] != rnk[y]) return rnk[x] < rnk[y];\n int rx = x + k < n ? rnk[x + k] : -1;\n int ry = y + k < n ? rnk[y + k] : -1;\n return rx < ry;\n };\n std::sort(sa.begin(), sa.end(), cmp);\n tmp[sa[0]] = 0;\n for (int i = 1; i < n; i++) {\n tmp[sa[i]] = tmp[sa[i - 1]] + (cmp(sa[i - 1], sa[i]) ? 1 : 0);\n }\n std::swap(tmp, rnk);\n }\n return sa;\n}\n// SA-IS, linear-time suffix array construction\n// Reference:\n// G. Nong, S. Zhang, and W. H. Chan,\n// Two Efficient Algorithms for Linear Time Suffix Array Construction\ntemplate \nstd::vector sa_is(const std::vector& s, int upper) {\n int n = int(s.size());\n if (n == 0) return {};\n if (n == 1) return {0};\n if (n == 2) {\n if (s[0] < s[1]) {\n return {0, 1};\n } else {\n return {1, 0};\n }\n }\n if (n < THRESHOLD_NAIVE) {\n return sa_naive(s);\n }\n if (n < THRESHOLD_DOUBLING) {\n return sa_doubling(s);\n }\n std::vector sa(n);\n std::vector ls(n);\n for (int i = n - 2; i >= 0; i--) {\n ls[i] = (s[i] == s[i + 1]) ? ls[i + 1] : (s[i] < s[i + 1]);\n }\n std::vector sum_l(upper + 1), sum_s(upper + 1);\n for (int i = 0; i < n; i++) {\n if (!ls[i]) {\n sum_s[s[i]]++;\n } else {\n sum_l[s[i] + 1]++;\n }\n }\n for (int i = 0; i <= upper; i++) {\n sum_s[i] += sum_l[i];\n if (i < upper) sum_l[i + 1] += sum_s[i];\n }\n auto induce = [&](const std::vector& lms) {\n std::fill(sa.begin(), sa.end(), -1);\n std::vector buf(upper + 1);\n std::copy(sum_s.begin(), sum_s.end(), buf.begin());\n for (auto d : lms) {\n if (d == n) continue;\n sa[buf[s[d]]++] = d;\n }\n std::copy(sum_l.begin(), sum_l.end(), buf.begin());\n sa[buf[s[n - 1]]++] = n - 1;\n for (int i = 0; i < n; i++) {\n int v = sa[i];\n if (v >= 1 && !ls[v - 1]) {\n sa[buf[s[v - 1]]++] = v - 1;\n }\n }\n std::copy(sum_l.begin(), sum_l.end(), buf.begin());\n for (int i = n - 1; i >= 0; i--) {\n int v = sa[i];\n if (v >= 1 && ls[v - 1]) {\n sa[--buf[s[v - 1] + 1]] = v - 1;\n }\n }\n };\n std::vector lms_map(n + 1, -1);\n int m = 0;\n for (int i = 1; i < n; i++) {\n if (!ls[i - 1] && ls[i]) {\n lms_map[i] = m++;\n }\n }\n std::vector lms;\n lms.reserve(m);\n for (int i = 1; i < n; i++) {\n if (!ls[i - 1] && ls[i]) {\n lms.push_back(i);\n }\n }\n induce(lms);\n if (m) {\n std::vector sorted_lms;\n sorted_lms.reserve(m);\n for (int v : sa) {\n if (lms_map[v] != -1) sorted_lms.push_back(v);\n }\n std::vector rec_s(m);\n int rec_upper = 0;\n rec_s[lms_map[sorted_lms[0]]] = 0;\n for (int i = 1; i < m; i++) {\n int l = sorted_lms[i - 1], r = sorted_lms[i];\n int end_l = (lms_map[l] + 1 < m) ? lms[lms_map[l] + 1] : n;\n int end_r = (lms_map[r] + 1 < m) ? lms[lms_map[r] + 1] : n;\n bool same = true;\n if (end_l - l != end_r - r) {\n same = false;\n } else {\n while (l < end_l) {\n if (s[l] != s[r]) {\n break;\n }\n l++;\n r++;\n }\n if (l == n || s[l] != s[r]) same = false;\n }\n if (!same) rec_upper++;\n rec_s[lms_map[sorted_lms[i]]] = rec_upper;\n }\n auto rec_sa =\n sa_is(rec_s, rec_upper);\n for (int i = 0; i < m; i++) {\n sorted_lms[i] = lms[rec_sa[i]];\n }\n induce(sorted_lms);\n }\n return sa;\n}\n} // namespace internal\nstd::vector suffix_array(const std::vector& s, int upper) {\n assert(0 <= upper);\n for (int d : s) {\n assert(0 <= d && d <= upper);\n }\n auto sa = internal::sa_is(s, upper);\n return sa;\n}\ntemplate std::vector suffix_array(const std::vector& s) {\n int n = int(s.size());\n std::vector idx(n);\n iota(idx.begin(), idx.end(), 0);\n sort(idx.begin(), idx.end(), [&](int l, int r) { return s[l] < s[r]; });\n std::vector s2(n);\n int now = 0;\n for (int i = 0; i < n; i++) {\n if (i && s[idx[i - 1]] != s[idx[i]]) now++;\n s2[idx[i]] = now;\n }\n return internal::sa_is(s2, now);\n}\nstd::vector suffix_array(const std::string& s) {\n int n = int(s.size());\n std::vector s2(n);\n for (int i = 0; i < n; i++) {\n s2[i] = s[i];\n }\n return internal::sa_is(s2, 255);\n}\n// Reference:\n// T. Kasai, G. Lee, H. Arimura, S. Arikawa, and K. Park,\n// Linear-Time Longest-Common-Prefix Computation in Suffix Arrays and Its\n// Applications\ntemplate \nstd::vector lcp_array(const std::vector& s,\n const std::vector& sa) {\n int n = int(s.size());\n assert(n >= 1);\n std::vector rnk(n);\n for (int i = 0; i < n; i++) {\n rnk[sa[i]] = i;\n }\n std::vector lcp(n - 1);\n int h = 0;\n for (int i = 0; i < n; i++) {\n if (h > 0) h--;\n if (rnk[i] == 0) continue;\n int j = sa[rnk[i] - 1];\n for (; j + h < n && i + h < n; h++) {\n if (s[j + h] != s[i + h]) break;\n }\n lcp[rnk[i] - 1] = h;\n }\n return lcp;\n}\nstd::vector lcp_array(const std::string& s, const std::vector& sa) {\n int n = int(s.size());\n std::vector s2(n);\n for (int i = 0; i < n; i++) {\n s2[i] = s[i];\n }\n return lcp_array(s2, sa);\n}\n// Reference:\n// D. Gusfield,\n// Algorithms on Strings, Trees, and Sequences: Computer Science and\n// Computational Biology\ntemplate std::vector z_algorithm(const std::vector& s) {\n int n = int(s.size());\n if (n == 0) return {};\n std::vector z(n);\n z[0] = 0;\n for (int i = 1, j = 0; i < n; i++) {\n int& k = z[i];\n k = (j + z[j] <= i) ? 0 : std::min(j + z[j] - i, z[i - j]);\n while (i + k < n && s[k] == s[i + k]) k++;\n if (j + z[j] < i + z[i]) j = i;\n }\n z[0] = n;\n return z;\n}\nstd::vector z_algorithm(const std::string& s) {\n int n = int(s.size());\n std::vector s2(n);\n for (int i = 0; i < n; i++) {\n s2[i] = s[i];\n }\n return z_algorithm(s2);\n}\n} // namespace atcoder\n#endif // ATCODER_STRING_HPP\n#ifndef ATCODER_TWOSAT_HPP\n#define ATCODER_TWOSAT_HPP 1\n#include \n#include \nnamespace atcoder {\n// Reference:\n// B. Aspvall, M. Plass, and R. Tarjan,\n// A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean\n// Formulas\nstruct two_sat {\n public:\n two_sat() : _n(0), scc(0) {}\n two_sat(int n) : _n(n), _answer(n), scc(2 * n) {}\n void add_clause(int i, bool f, int j, bool g) {\n assert(0 <= i && i < _n);\n assert(0 <= j && j < _n);\n scc.add_edge(2 * i + (f ? 0 : 1), 2 * j + (g ? 1 : 0));\n scc.add_edge(2 * j + (g ? 0 : 1), 2 * i + (f ? 1 : 0));\n }\n bool satisfiable() {\n auto id = scc.scc_ids().second;\n for (int i = 0; i < _n; i++) {\n if (id[2 * i] == id[2 * i + 1]) return false;\n _answer[i] = id[2 * i] < id[2 * i + 1];\n }\n return true;\n }\n std::vector answer() { return _answer; }\n private:\n int _n;\n std::vector _answer;\n internal::scc_graph scc;\n};\n} // namespace atcoder\n#endif // 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