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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<int, int> &v) const { return v.first * 31 + 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 = 101010; 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<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 ZERO(a) memset(a,0,sizeof(a)) #define MINUS(a) memset(a,0xff,sizeof(a)) #define INF(a) memset(a,0x3f3f3f3f3f3f3f3fLL,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 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
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(),"r",stdin); freopen(out.c_str(),"w",stdout);} void readf() {freopen("", "rt", stdin);} template <typename... T> void in(T &...a) { ((cin >> a), ...); } template<typename T> void readv(vector<T>& v){rep(i,0,sz(v)) in(v[i]);} template<typename T, typename U> void readp(pair<T,U>& A) {cin>>A.first>>A.second;} template<typename T, typename U> void readvp(vector<pair<T,U>>& A) {rep(i,0,sz(A)) readp(A[i]); } template<typename T> void readvall(vector<T>& v) {rep(i,0,sz(v)) rep(j,0,sz(v[i])) in(v[i][j]);} template<typename T> void readvv(vector<vector<T>>& A) {rep(i,0,sz(A)) readv(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) {}
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.matrix[i][j] = (res.matrix[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; }
ll get(ll y, ll x) { return matrix[y][x]; }
Matrix pow(ll n) { assert(r == c); Matrix res(r,r, 0,MOD); Matrix now = copy(); rep(i,0,r) res.matrix[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.matrix[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 &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 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 {1e9, 0}; if(x1 == x2) return {0, 1e9}; 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; if(!x) return 1; ll res = modpow(n,x>>1,MOD); res = (res * res) % MOD; if(x&1) res = (res * n) % MOD; return res; }
class Solution { ll uf[MAX_N]; ll find(ll u) { return uf[u] == u ? u : uf[u] = find(uf[u]); } void uni(ll u, ll v) { ll pu = find(u), pv = find(v); uf[pu] = uf[pv] = min(pu,pv); } public: bool canTraverseAllPairs(vector<int>& nums) { ASCEND(uf); unordered_map<ll,vll> mp; rep(i,0,sz(nums)) { for(ll j = 2; j * j <= nums[i]; j++) { if(nums[i] % j) continue; mp[j].push_back(i); while(nums[i] % j == 0) nums[i] /= j; } if(nums[i] != 1) mp[nums[i]].push_back(i); } for(auto& [k,vec] : mp) { rep(i,1,sz(vec)) uni(vec[0], vec[i]); } rep(i,0,sz(nums)) if(find(i)) return false; return true; } };
