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 #include <bits/stdc++.h>
#pragma optimization_level 3 #pragma GCC optimize("Ofast,nostackprotector,unrollloops,fastmath,O3") #pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx") #pragma GCC optimize("Ofast") #pragma GCC target("avx,avx2,fma") #pragma GCC optimization ("unrollloops")
using namespace std;
struct PairHash {inline std::size_t operator()(const std::pair<long long, long long> &v) const { return v.first * 31ll + v.second; }};
#define Code ios_base::sync_with_stdio(false); #define By ios::sync_with_stdio(0); #define Sumfi cout.tie(NULL);
using ll = long long; using ld = long double; using ull = unsigned long long;
const ld PI = acosl(1.0); const ll INF = 1e18; const ld EPS = 1e9; const ll MAX_N = 1010101; const ll mod = 1e9+7;
typedef pair<int,int> pii; typedef pair<ll,ll> pll; typedef vector<pll> vpll; typedef array<int,3> ai3; typedef array<ll,3> all3; typedef array<ll,4> all4; typedef array<ll,5> all5; typedef vector<all3> vall3; typedef vector<all4> vall4; typedef vector<all5> vall5; typedef pair<ld, ld> pld; typedef vector<pld> vpld; typedef vector<ld> vld; typedef vector<ll> vll; typedef vector<ull> vull; typedef vector<vll> vvll; typedef vector<int> vi; typedef vector<vi> vvi; typedef vector<bool> vb; typedef deque<ll> dqll; typedef deque<pll> dqpll; typedef pair<string, string> pss; typedef vector<pss> vpss; typedef vector<string> vs; typedef vector<vs> vvs; typedef unordered_set<ll> usll; typedef unordered_set<pll, PairHash> uspll; typedef unordered_map<ll, ll> umll; typedef unordered_map<pll, ll, PairHash> umpll;
#define precision(x) cout<<fixed;cout.precision(x); #define rep(i,m,n) for(ll i=m;i<n;i++) #define rrep(i,m,n) for(ll i=n;i>=m;i) #define all(a) begin(a), end(a) #define rall(a) rbegin(a), rend(a) #define uniq(a) sort(all(a)), a.erase(unique(all(a)),end(a)) #define ZERO(a) memset(a,0,sizeof(a)) #define MINUS(a) memset(a,0xff,sizeof(a)) #define INF(a) memset(a,0x3f3f3f3f3f3f3f3fLL,sizeof(a)) #define NEGINF(a) memset(a,0xcf,sizeof(a)) #define ASCEND(a) iota(all(a),0) #define sz(x) ll((x).size()) #define BIT(a,i) ((a>>i)&1) #define BITSHIFT(a,i,n) (((a<<i) & ((1ll<<n)  1))  (a>>(ni))) #define MAXBIT(a) (64ll  __builtin_clzll(a)  1ll) #define MINBIT(a) (__builtin_ctzll(a)) #define pyes cout<<"YES\n"; #define pno cout<<"NO\n"; #define endl "\n" #define pneg1 cout<<"1\n"; #define ppossible cout<<"possible\n"; #define pimpossible cout<<"impossible\n"; #define TC(x) cout<<"Case #"<<x<<": "; #define X first #define Y second
void __print(int x) { cerr << x; } void __print(long x) { cerr << x; } void __print(long long x) { cerr << x; } void __print(unsigned x) { cerr << x; } void __print(unsigned long x) { cerr << x; } void __print(unsigned long long x) { cerr << x; } void __print(float x) { cerr << x; } void __print(double x) { cerr << x; } void __print(long double x) { cerr << x; } void __print(char x) { cerr << '\'' << x << '\''; } void __print(const char *x) { cerr << '\"' << x << '\"'; } void __print(const string &x) { cerr << '\"' << x << '\"'; } void __print(bool x) { cerr << (x ? "true" : "false"); } template <typename A> void __print(const A &x); template <typename A, typename B> void __print(const pair<A, B> &p); template <typename... A> void __print(const tuple<A...> &t); template <typename T> void __print(stack<T> s); template <typename T> void __print(queue<T> q); template <typename T, typename... U> void __print(priority_queue<T, U...> q); template <typename A> void __print(const A &x) { bool first = true; cerr << '{'; for (const auto &i : x) { cerr << (first ? "" : ","), __print(i); first = false; } cerr << '}'; } template <typename A, typename B> void __print(const pair<A, B> &p) { cerr << '('; __print(p.first); cerr << ','; __print(p.second); cerr << ')'; } template <typename... A> void __print(const tuple<A...> &t) { bool first = true; cerr << '('; apply([&first](const auto &...args) { ((cerr << (first ? "" : ","), __print(args), first = false), ...); }, t); cerr << ')'; } template <typename T> void __print(stack<T> s) { vector<T> debugVector; while (!s.empty()) { T t = s.top(); debugVector.push_back(t); s.pop(); } reverse(debugVector.begin(), debugVector.end()); __print(debugVector); } template <typename T> void __print(queue<T> q) { vector<T> debugVector; while (!q.empty()) { T t = q.front(); debugVector.push_back(t); q.pop(); } __print(debugVector); } template <typename T, typename... U> void __print(priority_queue<T, U...> q) { vector<T> debugVector; while (!q.empty()) { T t = q.top(); debugVector.push_back(t); q.pop(); } __print(debugVector); } void _print() { cerr << "]\n"; } template <typename Head, typename... Tail> void _print(const Head &H, const Tail &...T) { __print(H); if (sizeof...(T)) cerr << ", "; _print(T...); } #ifndef ONLINE_JUDGE #define debug(...) cerr << "Line:" << __LINE__ << " [" << #__VA_ARGS__ << "] = ["; _print(__VA_ARGS__); #else #define debug(...) #endif
template <typename T> void print(T &&t) { cout << t << "\n"; } template<typename T> void printv(vector<T>v){ll n=v.size();rep(i,0,n){cout<<v[i];if(i+1!=n)cout<<' ';}cout<<endl;} template<typename T> void printvv(vector<vector<T>>v){ll n=v.size();rep(i,0,n)printv(v[i]);} template<typename T> void printvln(vector<T>v){ll n=v.size();rep(i,0,n)cout<<v[i]<<endl;} void fileIO(string in = "input.txt", string out = "output.txt") {freopen(in.c_str(),"ma",stdin); freopen(out.c_str(),"w",stdout);} void hackercupIO(string in) {fileIO("/Users/hayoungsong/Downloads/" + in, "/Users/hayoungsong/Downloads/solution.txt");} void readf() {freopen("", "rt", stdin);} template <typename... T> void in(T &...a) { ((cin >> a), ...); } template<typename T> void in(vector<T>& v){rep(i,0,sz(v)) in(v[i]);} template<typename T> void in(deque<T>& v){rep(i,0,sz(v)) in(v[i]);} template<typename T, typename U> void in(pair<T,U>& A) {in(A.first, A.second);} template<typename T, typename U> void in(vector<pair<T,U>>& A) {rep(i,0,sz(A)) in(A[i]); } template<typename T, std::size_t N> void in(vector<array<T,N>>& A) {rep(i,0,sz(A)) rep(j,0,sz(A[i])) in(A[i][j]); } template<typename T> void in(vector<vector<T>>& A) {rep(i,0,sz(A)) in(A[i]);}
struct Combination { vll fac, inv; ll n, MOD;
ll modpow(ll n, ll x, ll MOD = mod) { if(!x) return 1; ll res = modpow(n,x>>1,MOD); res = (res * res) % MOD; if(x&1) res = (res * n) % MOD; return res; }
Combination(ll _n, ll MOD = mod): n(_n + 1), MOD(MOD) { inv = fac = vll(n,1); rep(i,1,n) fac[i] = fac[i1] * i % MOD; inv[n  1] = modpow(fac[n  1], MOD  2, MOD); rrep(i,1,n  2) inv[i] = inv[i + 1] * (i + 1) % MOD; }
ll fact(ll n) {return fac[n];} ll nCr(ll n, ll r) { if(n < r or n < 0 or r < 0) return 0; return fac[n] * inv[r] % MOD * inv[nr] % MOD; } };
struct Matrix { ll r,c,MOD; vvll matrix; Matrix(ll r, ll c, ll v = 0, ll MOD = mod): r(r), c(c), matrix(vvll(r,vll(c,v))), MOD(MOD) {} Matrix(vvll m, ll MOD = mod) : r(sz(m)), c(sz(m[0])), matrix(m), MOD(MOD) {}
vector<ll>& operator[](ll pos) {return matrix[pos];} Matrix operator*(const Matrix& B) const { Matrix res(r, B.c, 0,MOD); rep(i,0,r) rep(j,0,B.c) rep(k,0,B.r) { res[i][j] = (res[i][j] + matrix[i][k] * B.matrix[k][j] % MOD) % MOD; } return res; }
Matrix copy() { Matrix copy(r,c,0,MOD); copy.matrix = matrix; return copy; }
Matrix pow(ll n) { assert(r == c); Matrix res(r,r, 0,MOD); Matrix now = copy(); rep(i,0,r) res[i][i] = 1; while(n) { if(n & 1) res = res * now; now = now * now; n /= 2; } return res; }
ll det() { if(r == 1) return matrix[0][0]; if(r == 2) return matrix[0][0] * matrix[1][1]  matrix[0][1] * matrix[1][0]; ll res = 0; rep(p,0,c) { Matrix mat(c1,c1); rep(i,1,r) rep(j,0,c) { if(j == p) continue; mat[i][j  (j >= p)] = matrix[i][j]; } res += matrix[0][p] * (p & 1 ? 1 : 1) * mat.det(); } return res; } };
template <typename T> struct Point { T y,x; Point(T y, T x) : y(y), x(x) {} Point(pair<T,T> p) : y(p.first), x(p.second) {} Point() {} void input() {cin>>y>>x;} friend ostream& operator<<(ostream& os, const Point<T>& p) { os<<p.y<<' '<<p.x<<'\n'; return os;} Point<T> operator+(Point<T>& p) {return Point<T>(y + p.y, x + p.x);} Point<T> operator(Point<T>& p) {return Point<T>(y  p.y, x  p.x);} Point<T> operator*(ll n) {return Point<T>(y*n,x*n); } Point<T> operator/(ll n) {return Point<T>(y/n,x/n); } bool operator==(const Point<T> &a) {return x == a.x && y == a.y;} bool operator!=(const Point<T> &a) {return x != a.x or y != a.y;} bool operator<(const Point &other) const {if (x == other.x) return y < other.y;return x < other.x;} Point<T> rotate(Point<T> center, ld angle) { ld si = sin(angle * PI / 180.), co = cos(angle * PI / 180.); ld y = this>y  center.y; ld x = this>x  center.x;
return Point<T>(y * co  x * si + center.y, y * si + x * co + center.x); } ld distance(Point<T> other) { T dy = abs(this>y  other.y); T dx = abs(this>x  other.x); return sqrt(dy * dy + dx * dx); }
T norm() { return x * x + y * y; } };
template<typename T> struct Line { Point<T> A, B; Line(Point<T> A, Point<T> B) : A(A), B(B) {} Line() {}
void input() { A = Point<T>(); B = Point<T>(); A.input(); B.input(); }
T ccw(Point<T> &a, Point<T> &b, Point<T> &c) { T res = a.x * b.y + b.x * c.y + c.x * a.y; res = (a.x * c.y + b.x * a.y + c.x * b.y); return res; }
bool on(Point<T> x) { return ccw(A,x,B) == 0; }
bool isIntersect(Line<T> o) { T p1p2 = ccw(A,B,o.A) * ccw(A,B,o.B); T p3p4 = ccw(o.A,o.B,A) * ccw(o.A,o.B,B); if (p1p2 == 0 && p3p4 == 0) { pair<T,T> p1(A.y, A.x), p2(B.y,B.x), p3(o.A.y, o.A.x), p4(o.B.y, o.B.x); if (p1 > p2) swap(p2, p1); if (p3 > p4) swap(p3, p4); return p3 <= p2 && p1 <= p4; } return p1p2 <= 0 && p3p4 <= 0; }
pair<bool,Point<ld>> intersection(Line<T> o) { if(!this>intersection(o)) return {false, {}}; ld det = 1. * (o.B.yo.A.y)*(B.xA.x)  1.*(o.B.xo.A.x)*(B.yA.y); ld t = ((o.B.xo.A.x)*(A.yo.A.y)  (o.B.yo.A.y)*(A.xo.A.x)) / det; return {true, {A.y + 1. * t * (B.y  A.y), B.x + 1. * t * (B.x  A.x)}}; }
pair<ld, ld> formula() { T y1 = A.y, y2 = B.y; T x1 = A.x, x2 = B.x; if(y1 == y2) return {0, (ld)y1}; if(x1 == x2) return {INF, (ld)x1}; ld a = 1. * (y2  y1) / (x2  x1); ld b = x1 * a + y1; return {a, b}; } };
template<typename T> struct Circle { Point<T> center; T radius; Circle(T y, T x, T radius) : center(Point<T>(y,x)), radius(radius) {} Circle(Point<T> center, T radius) : center(center), radius(radius) {} Circle() {}
void input() { center = Point<T>(); center.input(); cin>>radius; }
bool circumference(Point<T> p) { return (center.x  p.x) * (center.x  p.x) + (center.y  p.y) * (center.y  p.y) == radius * radius; }
bool intersect(Circle<T> c) { T d = (center.x  c.center.x) * (center.x  c.center.x) + (center.y  c.center.y) * (center.y  c.center.y); return (radius  c.radius) * (radius  c.radius) <= d and d <= (radius + c.radius) * (radius + c.radius); }
bool include(Circle<T> c) { T d = (center.x  c.center.x) * (center.x  c.center.x) + (center.y  c.center.y) * (center.y  c.center.y); return d <= radius * radius; } };
ll __gcd(ll x, ll y) { return !y ? x : __gcd(y, x % y); } all3 __exgcd(ll x, ll y) { if(!y) return {x,1,0}; auto [g,x1,y1] = __exgcd(y, x % y); return {g, y1, x1  (x/y) * y1}; } ll __lcm(ll x, ll y) { return x / __gcd(x,y) * y; } ll modpow(ll n, ll x, ll MOD = mod) {if(x<0){return modpow(modpow(n,x,MOD),MOD2,MOD);}n%=MOD;ll res=1;while(x){if(x&1){res=res*n%MOD;}n=n*n%MOD;x>>=1;}return res;} ll __xor(ll n) {return n%4==0?n:n%4==1?1:n%4==2?n+1:0;} ll __rangexor(ll l, ll r) {return __xor(r)^__xor(l1);}
class Solution { umll dp; ll helper(usll& us, ll x, ll d) { if(dp.count(x)) return dp[x]; ll u = ~x & ((1ll<<d)  1); queue<ll> q; usll vis; auto push = [&](ll x) { if(!vis.count(x)) { vis.insert(x); q.push(x); } }; push(u); ll res = d; while(res) { ll qsz = sz(q); while(qsz) { auto u = q.front(); q.pop(); if(us.count(u)) return dp[x] = res; rep(i,0,d) push(u ^ (1ll<<i)); } res; } return dp[x] = res; } public: vector<int> maxHammingDistances(vector<int>& nums, int m) { dp.clear(); usll us(all(nums)); vi res; rep(i,0,sz(nums)) { res.push_back(helper(us,nums[i],m)); } return res; } };
