//#include "worldmap.h"
//#include "triples.h"
//#include "festival.h"
//#include "migrations.h"
#include "obstacles.h"
#include <bits/stdc++.h>
#pragma GCC optimize("Ofast")
#pragma GCC target("sse2")
#define ll long long
#define db double
#define ld long double
#define endl '\n'
#define eb emplace_back
#define em emplace
#define pb push_back
#define pf push_front
#define pp pop_back
#define fr first
#define sc second
#define sz size
#define ir insert
#define yes cout << "YES" << endl
#define no cout << "NO" << endl
#define all(x) x.begin() , x.end()
#define alice cout << "Alice" << endl
#define bob cout << "Bob" << endl
#define fo(x , y) for(ll i = x;i < y;i++)
#define lli long long int
#define pii pair<lli , lli>
#define it int
using namespace std;
//mt19937 rd(time(NULL));
//IOI 2025 problems
const ll oo = 1LL << 60;
const ll mxn = 2e5+5;
const ll mxk = 1000001;
const ll md=1000000007;
//vector<int> g[mxn];
/* "souvenirs" -> 39 points for now
void buy_souvenirs(int N, ll P0) {
if(N==2){transaction(P0-1) ;return;}
if(N==3){
auto x=transaction(P0-1);
if(x.fr.sz()==1){ll vl=P0-1-x.sc; transaction(vl-1);transaction(vl-1);}
else{ll vl= (P0-1-x.sc) ; vl+=(2-vl%2)%2;vl/=2; transaction(vl-1);}
return;
}
vector<ll>cn(N,0) , vl(N);
vl[0]=P0;
for(int i =1;i < N;i++) {
auto x =transaction(vl[i-1] - 1) ;
for(int y: x.fr) cn[y]++;
if(x.fr.sz() ==1) vl[i] = vl[i-1] - 1 -x.sc;
else vl[i] = vl[i-1] - 2;
}
for(int i = 1; i<N;i++) {
while(cn[i] <i) {
transaction(vl[i]);
cn[i]++;
}
}
}
*/
/* "World Map" -> 72 points for now
vector<vector<int>> regulate(vector<vector<int>> board) {
int n= board.sz();
int m=board[0].sz() ;
int k = max(n,m);
vector an=vector(k,vector<int> (k));
for(int i=0;i<n;i++)for(int j = 0;j<m ;j++)an[i][j]=board[i][j];
for(int i=0;i<k;i++){for(int j=0;j<k;j++){
if(!an[i][j]){
if(j>0&&an[i][j-1])an[i][j] =an[i][j-1];
else if(i>0&&an[i-1][j]) an[i][j] =an[i-1][j];
else assert(false);
}
}}
return an;
}
bool vs[mxn] ;
vector<pair<int, bool>> st;
void dfs(int v, int pa=-1) {
vs[v]=true;
for(int u:g[v]) {
if(vs[u])continue;
st.pb({u,true});
dfs(u,v);
st.pb({v,false});
}
}
vector<vector<int>> create_map(int N, int M , vector<int> A , vector<int> B) {
for(int i = 1 ; i <= N ;i++) g[i].clear();
fill(vs,vs + N+1, false) ;
for(int i = 0 ;i < M;i++){g[A[i]].pb(B[i]) ;g[B[i]].pb(A[i]);}
st=vector<pair<int,bool>>{{1,true}};
dfs(1);
vector ans=vector(N*3 +(N-1), vector<int> (N*2)) ;
int x= 0;
for(auto [v,fi]:st) {
if(fi) {
for(int j=0;j<3;j++){for(int k = 0; k < N*2;k++)ans[x+j][k]=v;}
for(int j =0;j <g[v].sz() ;j++)ans[x+1][j*2] = g[v][j];
x+=3;
}
else{
for(int k =0;k<N*2;k++)ans[x][k]=v;
x++;
}
}
return regulate(ans) ;
}
*/
//"Triple Peaks" , goal -> 50 , result -> 56.29
/*set<tuple<int, int , int>> sn;
inline bool works(int i , int j , int k , const vector<int> &H){
int p[3] = {i , j , k};
sort(p , p +3);
i = p[0] , j = p[1] , k =p[2];
if(0>i||i >=j || j >=k || k >=H.sz()) return false;
if(j-i == H[k] && k - j == H[i]) return false;
if(sn.count({i,j,k})) return false;
int d[3] = {j - i , k - j , k -i};
int h[3] = {H[i] , H[j] ,H[k]} ;
sort(d , d+3);
sort(h, h+3);
if(d[0]==h[0] && d[1] == h[1] && d[2] == h[2]) {sn.em(i , j , k); return true;}
return false;
}
struct comb{
map<int, int> a;
ll operator*(const comb& b)const{
if(a.sz() > b.a.sz()) return b * *this;
ll o=0;
for(auto [i,v] :a) if(b.a.count(i)) o+=(ll)v*b.a.at(i);
return o;
}
void insert(int v){a[v]++;}
void remove(int v) {if(!--a[v]){a.erase(v);}}
};
ll count_triples(vector<int> H) {
cerr << "count_triples(...)\n";
const int N = H.sz();
ll ct=0;
vector<int> xm(N) , xp(N);
for(int x = 0 ; x< N ; x++) {xm[x]=x-H[x]; xp[x] =x+H[x];}
map<int, comb>lf, rt;
for(int i =0 ; i < N;i++)rt[xp[i]].insert(xm[i]);
for(int j = 0 ; j < N;j++){
rt[xp[j]].remove(xm[j]);
ct += lf[xm[j]] * rt[xp[j]];
lf[xm[j]].insert(xp[j]);
}
cerr<<"pre-trivial: "<< ct<< endl;
for(int i =0 ; i <N;i++){
for(int j :{i + H[i], i-H[i]}){
if(j < 0||j>=N)continue;
for(int k :{i+H[j], i-H[j], j+H[j],j-H[j]}) ct+=works(i , j , k , H);
}
}
cerr<< " = "<<ct<<endl;
return ct;
}
vector<int> construct_range(int M , int K) {
return {};
}*/
// CF easy problem
/*
ll n ,m , t;
ll c[mxn] , z[mxn] , h[mxn] , f[mxn] , a[mxn];
vector<ll> e[mxn];
void css(ll nd , ll pr = -1) {
z[nd] = 1 ;
for(auto ch:e[nd]){if(ch!=pr) css(ch,nd) ,z[nd]+=z[ch];}
ll mxs = -1;
for(auto ch:e[nd]) {if(ch!=pr && z[ch] > mxs) mxs = z[h[nd] =ch];}
}
void ubdf(ll vl){
if(++f[vl] == m) t += vl;
if(f[vl] > m)m = f[t = vl];
}
void pch(ll nd , ll pr = -1) {
ubdf(c[nd]) ;
for(auto ch: e[nd]){if(ch == pr) continue; pch(ch ,nd);}
}
void fc(ll nd , ll pr =-1, bool ir = false) {
if(!h[nd]) {
a[nd] = c[nd] ;
if(ir) ubdf(c[nd]) ;
return;
}
for(auto ch :e[nd]) {if(ch == pr|| ch == h[nd])continue; fc(ch,nd) ;}
if(h[nd])fc(h[nd] , nd , true) ;
for(auto ch :e[nd]) {if(ch == pr|| ch == h[nd])continue; pch(ch ,nd);}
ubdf(c[nd]);
a[nd] = t;
if(!ir) {for(ll i = 1 ; i<= n ;i++){f[i] = 0;} m = t = 0;}
}*/
/*ll in() {
ll a =0;
cin >>a ;
return a;
}
struct bs{
ll b[20] ;
ll p[20] ;
void ins(ll x , ll pp){
for(ll i =19; ~i; i--){
if(x >> i & 1){
ll &k=b[i];
if(!k && (p[i] = pp ,k=x , 1))return;
if(p[i] < pp){
pp^=p[i] ^=pp^=p[i];
k^=x^=k^=x;
}
x^=k;
}
}
}
ll qry(ll l) {
ll a=0;
for(ll i = 19 ; ~i;i--){
if((~a >> i & 1) && p[i] >=l) a ^= b[i];
}
return a;
}
}bs[mxn];*/
// IOI 2025, day 2
// Task 1 : goal -> 50 points , my own result -> 63
/*vector<it> max_coupons(it A , vector<it> P , vector<it> T) {
auto comp=[&](it u,it v){
ll cv=1ll*P[u]*T[u]*T[v]+1ll*P[v]*T[v];
ll cu=1ll*P[v]*T[v]*T[u]+1ll*P[u]*T[u];
return cv<cu;
};
auto buy=[&](vector<it>& ids){
sort(begin(ids),end(ids),comp);
ll hs=A;
for(it i:ids){
hs=min((hs-P[i])*T[i],oo);
if(hs<0) return hs;
}
return hs;
};
vector<pair<it,it>> a[4]; it n=(it)P.size();
for(it i=0;i<n;i++)a[T[i]-1].pb({P[i],i});
for(it i=0;i<4;i++)sort(begin(a[i]),end(a[i]));
vector<ll> s1={0};
for(auto [x,_]:a[0]) s1.pb(s1.back()+x);
vector<it> ids(n); iota(begin(ids),end(ids),0);
if(buy(ids)>=0)return ids;
it bs=-1; vector<it> ans;
for(it i=0;i<=min(it(a[1].size()),65);i++)
for(it j=0;j<=min(it(a[2].size()),45);j++)
for(it k=0;k<=min(it(a[3].size()),35);k++){
vector<it> ids;
for(it u=0;u<i;u++)ids.pb(a[1][u].sc);
for(it u=0;u<j;u++)ids.pb(a[2][u].sc);
for(it u=0;u<k;u++)ids.pb(a[3][u].sc);
ll hs=buy(ids);
if(hs<0)continue;
it u=upper_bound(begin(s1),end(s1),hs)-begin(s1)-1;
it szi=(it)ids.size();
if(u+szi>bs){bs=u+szi; ans=ids;for(it q=0;q<u;q++) ans.pb(a[0][q].sc);}
}
return ans;
}*/
//Task 2 : goal -> 40 , result -> 30 ):
/*const it rf[7]={9999,9998,9997,9996,9995,9994,9993};
const it pw[7]={1,4,16,64,256,1024,4096};
vector<it> dep,par;
it send_message(it _N,it i,it Pi) {
if(i==1) {dep.assign(_N,0); par.assign(_N,0);}
par[i]=Pi; dep[i]=dep[par[i]]+1;it res=0;
for(it k=0;k<=6;k++){
if(i==::rf[k]) {
it bs=-1,id=-1;
for(it j=0;j<_N;j++)if(dep[j]>bs){bs=dep[j];id=j;}
if(id==i)res=4;
else {for(it j=6;j>k;j--)id%=pw[j];res=id/pw[k];}
}
}
return res;
}
pair<it,it> longest_path(vector<it> S) {
it v=-1;
for(it k=0;k<=6;k++)if(S[::rf[k]]==4) {v=::rf[k];break;}
if(v==-1) {
it res=0;
for(it k=0;k<=6;k++)res+=pw[k]*S[::rf[k]];
v=res;
}
return {0,v};
}*/
// Task 3 : goal -> 50
it n,m, bj[5][mxn];
it dx[]={1,-1,0,0}, dy[]={0,0,1,-1};
bool mz[5][mxn];
void dfs(it x,it y,it c) {
if(x<0 or x>=n or y<0 or y>=m)return;
if(bj[x][y]!=0 or !mz[x][y])return;
bj[x][y]=c;
for(it i=0;i<4;i++)dfs(x+dx[i],y+dy[i],c);
}
void initialize(vector<it> T,vector<it> H) {
if(T.sz()>3) {
vector<it> nt;
nt.pb(T[0]);
nt.pb(T[1]);
nt.pb(T.back());
T=nt;
}
n=T.sz() , m=H.sz();
for(it i=0;i<n;i++)for(it j=0;j<m;j++)mz[i][j]=T[i]>H[j];
it cnt=0;
for(it i=0;i<n;i++)for(it j=0;j<m;j++)dfs(i,j,++cnt);
return;
}
bool can_reach(it L,it R,it S,it D){return bj[0][S]==bj[0][D];}
