//https://atcoder.jp/contests/arc123/submissions/24171616 #include using namespace std; using ll=long long; #define int ll #define rng(i,a,b) for(int i=int(a);i=int(a);i--) #define per(i,b) gnr(i,0,b) #define pb push_back #define eb emplace_back #define a first #define b second #define bg begin() #define ed end() #define all(x) x.bg,x.ed #define si(x) int(x.size()) #ifdef LOCAL #define dmp(x) cerr<<__LINE__<<" "<<#x<<" "< bool chmax(t&a,u b){if(a bool chmin(t&a,u b){if(b using vc=vector; template using vvc=vc>; using pi=pair; using vi=vc; template ostream& operator<<(ostream& os,const pair& p){ return os<<"{"< ostream& operator<<(ostream& os,const vc& v){ os<<"{"; for(auto e:v)os< void dmpr(ostream&os,const T&t,const Args&... args){ os< ostream& operator<<(ostream&os,const array&a){ return os<(all(a)); } template void print_tuple(ostream&,const T&){ } template void print_tuple(ostream&os,const T&t){ if(i)os<<","; os<(t); print_tuple(os,t); } template ostream& operator<<(ostream&os,const tuple&t){ os<<"{"; print_tuple<0,tuple,Args...>(os,t); return os<<"}"; } template void print(t x,int suc=1){ cout<>i; return i; } vi readvi(int n,int off=0){ vi v(n); rep(i,n)v[i]=read()+off; return v; } pi readpi(int off=0){ int a,b;cin>>a>>b; return pi(a+off,b+off); } template void print(const pair&p,int suc=1){ print(p.a,2); print(p.b,suc); } template void print(const vector&v,int suc=1){ rep(i,v.size()) print(v[i],i==int(v.size())-1?suc:2); } string readString(){ string s; cin>>s; return s; } template T sq(const T& t){ return t*t; } //#define CAPITAL void yes(bool ex=true){ #ifdef CAPITAL cout<<"YES"<<"\n"; #else cout<<"Yes"<<"\n"; #endif if(ex)exit(0); #ifdef LOCAL cout.flush(); #endif } void no(bool ex=true){ #ifdef CAPITAL cout<<"NO"<<"\n"; #else cout<<"No"<<"\n"; #endif if(ex)exit(0); #ifdef LOCAL cout.flush(); #endif } void possible(bool ex=true){ #ifdef CAPITAL cout<<"POSSIBLE"<<"\n"; #else cout<<"Possible"<<"\n"; #endif if(ex)exit(0); #ifdef LOCAL cout.flush(); #endif } void impossible(bool ex=true){ #ifdef CAPITAL cout<<"IMPOSSIBLE"<<"\n"; #else cout<<"Impossible"<<"\n"; #endif if(ex)exit(0); #ifdef LOCAL cout.flush(); #endif } constexpr ll ten(int n){ return n==0?1:ten(n-1)*10; } const ll infLL=LLONG_MAX/3; #ifdef int const int inf=infLL; #else const int inf=INT_MAX/2-100; #endif 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 botbit(signed a){ return a==0?32:__builtin_ctz(a); } int botbit(ll a){ return a==0?64:__builtin_ctzll(a); } int popcount(signed t){ return __builtin_popcount(t); } int popcount(ll t){ return __builtin_popcountll(t); } bool ispow2(int i){ return i&&(i&-i)==i; } ll mask(int i){ return (ll(1)< void mkuni(vc&v){ sort(all(v)); v.erase(unique(all(v)),v.ed); } ll rand_int(ll l, ll r) { //[l, r] #ifdef LOCAL static mt19937_64 gen; #else static mt19937_64 gen(chrono::steady_clock::now().time_since_epoch().count()); #endif return uniform_int_distribution(l, r)(gen); } template void myshuffle(vc&a){ rep(i,si(a))swap(a[i],a[rand_int(0,i)]); } template int lwb(const vc&v,const t&a){ return lower_bound(all(v),a)-v.bg; } vvc readGraph(int n,int m){ vvc g(n); rep(i,m){ int a,b; cin>>a>>b; //sc.read(a,b); a--;b--; g[a].pb(b); g[b].pb(a); } return g; } vvc readTree(int n){ return readGraph(n,n-1); } template t extgcd(t a,t b,t&x,t&y){ if(b==0){ x=1; y=0; return a; }else{ t g=extgcd(b,a%b,y,x); y-=a/b*x; return g; } } const int vmax=ten(6)+10; bool pri[vmax]; vi ps; int sf[vmax]; //smallest factor, sf[1] is undefined(0) void linear_sieve(){ rng(i,2,vmax) pri[i]=1; rng(i,2,vmax){ if(pri[i]){ ps.pb(i); sf[i]=i; } for(int j=0;i*ps[j] 0 && a % b); } bool cmpf(pi a,pi b){ return a.a*b.b((num*v+den)/den,0); int s=0; for(auto m:ms){ if(v*m>n)break; s+=mu[m]; } ans+=w*s; } dmp2(n,num,den,ans); return ans; } //num/den < p/q, (0<=p,1<=q<=n) //の形で表される既約分数を小さい方から len 個返す //O(N+Len) vc enum_frac(int len,int n,int num,int den){ vc res; res.reserve(len); auto dfs=[&](auto self,pi l,pi r)->void{ if(si(res)==len)return; pi m(l.a+r.a,l.b+r.b); if(m.b>n)return; self(self,l,m); if(si(res)==len)return; res.pb(m); self(self,m,r); }; pi l(fdiv(num,den),1); pi r(l.a+1,1); rng(v,2,n+1){ pi w(fdiv(num*v,den),v); if(cmpf(l,w))l=w; w.a++; if(cmpf(w,r))r=w; } while(si(res)r.b){ l.a-=r.a; l.b-=r.b; }else{ assert(l.b==1); l.a++; r.a++; res.pb(l); dfs(dfs,l,r); } } return res; } /* int gcd(int a,int b){ if(a<0)a=-a; if(b<0)b=-b; if(a==0)return b; if(b==0)return a; int s=botbit(a|b); a>>=botbit(a); do{ b>>=botbit(b); if(a>b)swap(a,b); b-=a; }while(b); return a< enum_frac(int len,int n,int num,int den){ struct picmp{ bool operator()(const pi&a,const pi&b)const{ return a.a*b.b>a.b*b.a; } }; priority_queue,picmp> pq; rng(v,1,n+1){ int w=fdiv(num*v,den); do{w++; }while(gcd(w,v)!=1); pq.emplace(w,v); } vc res; while(si(res)>mx>>l; r=l+1; if(phi[mx]==l){ cout<<"1/1\n"; }else if(phi[mx]>l){ int x,y; int g=extgcd(a,b,x,y); dmp2(a,b,g); a/=g; b/=g; mx/=g; if(y>0){ x+=b; y-=a; } assert(y<=0); y=-y; int den=sq(mx+1); int num=cdiv(y*den,a)-1; dmp2(num,den); { int tmp=count_frac(mx,num,den); l+=tmp; r+=tmp; } dmp2(l,r); int lw=-1,up=den; while(up-lw>1){ const int mid=(lw+up)/2; int cnt=count_frac(mx,mid,den); if(cnt<=l)lw=mid; else up=mid; } dmp2(up,den); dmp2(r-l,mx,lw,den); auto ans=enum_frac(r-l,mx,lw,den); dmp(ans); vi res; for(auto [p,q]:ans){ //res.pb(q*g); cout<phi[mx]){ cout<<-1<<"\n"; }else{ l=phi[mx]+1-l; r=l+1; int x,y; int g=extgcd(a,b,x,y); dmp2(a,b,g); a/=g; b/=g; mx/=g; if(y>0){ x+=b; y-=a; } assert(y<=0); y=-y; int den=sq(mx+1); int num=cdiv(y*den,a)-1; dmp2(num,den); { int tmp=count_frac(mx,num,den); l+=tmp; r+=tmp; } dmp2(l,r); int lw=-1,up=den; while(up-lw>1){ const int mid=(lw+up)/2; int cnt=count_frac(mx,mid,den); if(cnt<=l)lw=mid; else up=mid; } dmp2(up,den); dmp2(r-l,mx,lw,den); auto ans=enum_frac(r-l,mx,lw,den); dmp(ans); vi res; for(auto [p,q]:ans){ //res.pb(q*g); cout<>t; rep(_,t) slv(); }