#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; namespace { using Integer = long long; //__int128; template istream& operator >> (istream& is, pair& p){return is >> p.first >> p.second;} template istream& operator >> (istream& is, vector& vec){for(T& val: vec) is >> val; return is;} template istream& operator , (istream& is, T& val){ return is >> val;} template ostream& operator << (ostream& os, const pair& p){return os << p.first << " " << p.second;} template ostream& operator << (ostream& os, const vector& vec){for(size_t i=0; i ostream& operator , (ostream& os, const T& val){ return os << " " << val;} template void print(const H& head){ cout << head; } template void print(const H& head, const T& ... tail){ cout << head << " "; print(tail...); } template void println(const T& ... values){ print(values...); cout << endl; } template void eprint(const H& head){ cerr << head; } template void eprint(const H& head, const T& ... tail){ cerr << head << " "; eprint(tail...); } template void eprintln(const T& ... values){ eprint(values...); cerr << endl; } class range{ Integer start_, end_, step_; public: struct range_iterator{ Integer val, step_; range_iterator(Integer v, Integer step) : val(v), step_(step) {} Integer operator * (){return val;} void operator ++ (){val += step_;} bool operator != (range_iterator& x){return step_ > 0 ? val < x.val : val > x.val;} }; range(Integer len) : start_(0), end_(len), step_(1) {} range(Integer start, Integer end) : start_(start), end_(end), step_(1) {} range(Integer start, Integer end, Integer step) : start_(start), end_(end), step_(step) {} range_iterator begin(){ return range_iterator(start_, step_); } range_iterator end(){ return range_iterator( end_, step_); } }; inline string operator "" _s (const char* str, size_t size){ return move(string(str)); } constexpr Integer my_pow(Integer x, Integer k, Integer z=1){return k==0 ? z : k==1 ? z*x : (k&1) ? my_pow(x*x,k>>1,z*x) : my_pow(x*x,k>>1,z);} constexpr Integer my_pow_mod(Integer x, Integer k, Integer M, Integer z=1){return k==0 ? z%M : k==1 ? z*x%M : (k&1) ? my_pow_mod(x*x%M,k>>1,M,z*x%M) : my_pow_mod(x*x%M,k>>1,M,z);} constexpr unsigned long long operator "" _ten (unsigned long long value){ return my_pow(10,value); } inline int k_bit(Integer x, int k){return (x>>k)&1;} //0-indexed mt19937 mt(chrono::duration_cast(chrono::steady_clock::now().time_since_epoch()).count()); template string join(const vector& v, const string& sep){ stringstream ss; for(size_t i=0; i0) ss << sep; ss << v[i]; } return ss.str(); } inline string operator * (string s, int k){ string ret; while(k){ if(k&1) ret += s; s += s; k >>= 1; } return ret; } } constexpr long long mod = 9_ten + 7; using real = long double; //A[n*p]*b[p*1] = x[n*1] vector > gaussian_elimination(vector > &A, vector > &x){ bool valid = true; int rank=0; int n = A.size(); int p = A[0].size(); vector< vector > R = A; for(int i=0; i=0; i--){ for(int j=i-1; j>=0; j--){ real c = R[j][i]; for(int k=i; k> n,m; assert(m != n-1); vector c(n, 0); for(int i=1; i> c[i]; } vector> M(n, vector(n, 0.0)); for(auto i : range(n) ){ if(i n-1 ){ p = n-1-(p-n+1); } M[i][p] += (real)1.0/m; } if(i==n-1){ M[i][i] = 1.0; } } vector> I(n-1, vector(n-1, 0.0)); for(auto i : range(n-1) ){ I[i][i] = 1.0; } vector> Q(n-1, vector(n-1, 0.0)); for(auto i : range(n-1) ){ for(auto j : range(n-1) ){ Q[i][j] = I[i][j] - M[i][j]; } } auto P_ = gaussian_elimination(Q,I); vector> P(n-1, vector(n-1, 0.0)); for(auto i : range(n-1) ){ for(auto j : range(n-1) ){ P[i][j] = P_[i][j+n-1]; } } //eprintln(join(P, "\n")); vector EE(n-1, 0); for(int i : range(n-1)){ //EE[i] -= c[i]; for(int j : range(0, n-1)){ EE[i] += P[i][j] * c[j]; } } EE.push_back(0.0); //eprintln("exp : " , join(EE, " ")); real ans = EE[0]; for(int i=1; i<=m; i++){ ans = min(ans, EE[i]); } //eprintln(ans); for(int p=m+1; p> ii(p, vector(p, 0.0)); vector> qq(p, vector(p, 0.0)); for(auto i : range(p) ){ ii[i][i] = 1.0; qq[i][i] = 1.0; } for(int i=0; i