#include "towers.h"
#ifndef LOCAL
#pragma GCC optimize ("Ofast")
#pragma GCC optimize ("unroll-loops")
#endif
#include <bits/stdc++.h>
using namespace std;
using ll=long long;
//#define int ll
#define rng(i,a,b) for(int i=int(a);i<int(b);i++)
#define rep(i,b) rng(i,0,b)
#define gnr(i,a,b) for(int i=int(b)-1;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<<" "<<x<<endl
#else
#define dmp(x) void(0)
#endif
template<class t,class u> bool chmax(t&a,u b){if(a<b){a=b;return true;}else return false;}
template<class t,class u> bool chmin(t&a,u b){if(b<a){a=b;return true;}else return false;}
template<class t> using vc=vector<t>;
template<class t> using vvc=vc<vc<t>>;
using pi=pair<int,int>;
using vi=vc<int>;
template<class t,class u>
ostream& operator<<(ostream& os,const pair<t,u>& p){
return os<<"{"<<p.a<<","<<p.b<<"}";
}
template<class t> ostream& operator<<(ostream& os,const vc<t>& v){
os<<"{";
for(auto e:v)os<<e<<",";
return os<<"}";
}
#define mp make_pair
#define mt make_tuple
#define one(x) memset(x,-1,sizeof(x))
#define zero(x) memset(x,0,sizeof(x))
#ifdef LOCAL
void dmpr(ostream&os){os<<endl;}
template<class T,class... Args>
void dmpr(ostream&os,const T&t,const Args&... args){
os<<t<<" ";
dmpr(os,args...);
}
#define dmp2(...) dmpr(cerr,__LINE__,##__VA_ARGS__)
#else
#define dmp2(...) void(0)
#endif
using uint=unsigned;
using ull=unsigned long long;
template<class t,size_t n>
ostream& operator<<(ostream&os,const array<t,n>&a){
return os<<vc<t>(all(a));
}
template<int i,class T>
void print_tuple(ostream&,const T&){
}
template<int i,class T,class H,class ...Args>
void print_tuple(ostream&os,const T&t){
if(i)os<<",";
os<<get<i>(t);
print_tuple<i+1,T,Args...>(os,t);
}
template<class ...Args>
ostream& operator<<(ostream&os,const tuple<Args...>&t){
os<<"{";
print_tuple<0,tuple<Args...>,Args...>(os,t);
return os<<"}";
}
ll read(){
ll i;
cin>>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<class t>
void print_single(t x,int suc=1){
cout<<x;
if(suc==1)
cout<<"\n";
if(suc==2)
cout<<" ";
}
template<class t,class u>
void print_single(const pair<t,u>&p,int suc=1){
print_single(p.a,2);
print_single(p.b,suc);
}
template<class T>
void print_single(const vector<T>&v,int suc=1){
rep(i,v.size())
print_single(v[i],i==int(v.size())-1?suc:2);
}
template<class T>
void print_offset(const vector<T>&v,ll off,int suc=1){
rep(i,v.size())
print_single(v[i]+off,i==int(v.size())-1?suc:2);
}
template<class T,size_t N>
void print_single(const array<T,N>&v,int suc=1){
rep(i,N)
print_single(v[i],i==int(N)-1?suc:2);
}
template<class T>
void print(const T&t){
print_single(t);
}
template<class T,class ...Args>
void print(const T&t,const Args&...args){
print_single(t,2);
print(args...);
}
string readString(){
string s;
cin>>s;
return s;
}
template<class T>
T sq(const T& t){
return t*t;
}
void YES(bool ex=true){
cout<<"YES\n";
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}
void NO(bool ex=true){
cout<<"NO\n";
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}
void Yes(bool ex=true){
cout<<"Yes\n";
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}
void No(bool ex=true){
cout<<"No\n";
if(ex)exit(0);
#ifdef LOCAL
cout.flush();
#endif
}
//#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 topbit(ull 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 botbit(ull a){
return a==0?64:__builtin_ctzll(a);
}
int popcount(signed t){
return __builtin_popcount(t);
}
int popcount(ll t){
return __builtin_popcountll(t);
}
int popcount(ull t){
return __builtin_popcountll(t);
}
int bitparity(ll t){
return __builtin_parityll(t);
}
bool ispow2(int i){
return i&&(i&-i)==i;
}
ll mask(int i){
return (ll(1)<<i)-1;
}
ull umask(int i){
return (ull(1)<<i)-1;
}
ll minp2(ll n){
if(n<=1)return 1;
else return ll(1)<<(topbit(n-1)+1);
}
bool inc(int a,int b,int c){
return a<=b&&b<=c;
}
template<class t> void mkuni(vc<t>&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<ll>(l, r)(gen);
}
ll rand_int(ll k){ //[0,k)
return rand_int(0,k-1);
}
template<class t>
void myshuffle(vc<t>&a){
rep(i,si(a))swap(a[i],a[rand_int(0,i)]);
}
template<class t>
int lwb(const vc<t>&v,const t&a){
return lower_bound(all(v),a)-v.bg;
}
template<class t>
bool bis(const vc<t>&v,const t&a){
return binary_search(all(v),a);
}
vvc<int> readGraph(int n,int m){
vvc<int> 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<int> readTree(int n){
return readGraph(n,n-1);
}
template<class t>
vc<t> presum(const vc<t>&a){
vc<t> s(si(a)+1);
rep(i,si(a))s[i+1]=s[i]+a[i];
return s;
}
vc<ll> presum(const vi&a){
vc<ll> s(si(a)+1);
rep(i,si(a))s[i+1]=s[i]+a[i];
return s;
}
//BIT で数列を管理するときに使う (CF850C)
template<class t>
vc<t> predif(vc<t> a){
gnr(i,1,si(a))a[i]-=a[i-1];
return a;
}
template<class t>
vvc<ll> imos(const vvc<t>&a){
int n=si(a),m=si(a[0]);
vvc<ll> b(n+1,vc<ll>(m+1));
rep(i,n)rep(j,m)
b[i+1][j+1]=b[i+1][j]+b[i][j+1]-b[i][j]+a[i][j];
return b;
}
//verify してないや
void transvvc(int&n,int&m){
swap(n,m);
}
template<class t,class... Args>
void transvvc(int&n,int&m,vvc<t>&a,Args&...args){
assert(si(a)==n);
vvc<t> b(m,vi(n));
rep(i,n){
assert(si(a[i])==m);
rep(j,m)b[j][i]=a[i][j];
}
a.swap(b);
transvvc(n,m,args...);
}
//CF854E
void rotvvc(int&n,int&m){
swap(n,m);
}
template<class t,class... Args>
void rotvvc(int&n,int&m,vvc<t>&a,Args&...args){
assert(si(a)==n);
vvc<t> b(m,vi(n));
rep(i,n){
assert(si(a[i])==m);
rep(j,m)b[m-1-j][i]=a[i][j];
}
a.swap(b);
rotvvc(n,m,args...);
}
//ソートして i 番目が idx[i]
//CF850C
template<class t>
vi sortidx(const vc<t>&a){
int n=si(a);
vi idx(n);iota(all(idx),0);
sort(all(idx),[&](int i,int j){return a[i]<a[j];});
return idx;
}
//vs[i]=a[idx[i]]
//例えば sortidx で得た idx を使えば単にソート列になって返ってくる
//CF850C
template<class t>
vc<t> a_idx(const vc<t>&a,const vi&idx){
int n=si(a);
assert(si(idx)==n);
vc<t> vs(n);
rep(i,n)vs[i]=a[idx[i]];
return vs;
}
//CF850C
vi invperm(const vi&p){
int n=si(p);
vi q(n);
rep(i,n)q[p[i]]=i;
return q;
}
template<class t,class s=t>
s SUM(const vc<t>&a){
return accumulate(all(a),s(0));
}
template<class t>
t MAX(const vc<t>&a){
return *max_element(all(a));
}
template<class t>
pair<t,int> MAXi(const vc<t>&a){
auto itr=max_element(all(a));
return mp(*itr,itr-a.bg);
}
template<class t>
t MIN(const vc<t>&a){
return *min_element(all(a));
}
template<class t>
pair<t,int> MINi(const vc<t>&a){
auto itr=min_element(all(a));
return mp(*itr,itr-a.bg);
}
template<class t,class u>
pair<t,u> operator+(const pair<t,u>&a,const pair<t,u>&b){
return mp(a.a+b.a,a.b+b.b);
}
vi vid(int n){
vi res(n);iota(all(res),0);
return res;
}
template<class S>
S revv(S s){
reverse(all(s));
return s;
}
pi operator+(pi a,pi b){return pi(a.a+b.a,a.b+b.b);}
template<class t>
t gpp(vc<t>&vs){
assert(si(vs));
t res=move(vs.back());
vs.pop_back();
return res;
}
template<class t>
void pb(vc<t>&a,const vc<t>&b){
a.insert(a.ed,all(b));
}
template<class t>
vc<t> cat(vc<t> a,const vc<t>&b){
pb(a,b);
return a;
}
template<class t,class u>
vc<t>& operator+=(vc<t>&a,u x){
for(auto&v:a)v+=x;
return a;
}
template<class t,class u>
vc<t> operator+(vc<t> a,u x){
return a+=x;
}
template<class t,class u>
vc<t>& operator-=(vc<t>&a,u x){
for(auto&v:a)v-=x;
return a;
}
template<class t,class u>
vc<t>& operator-(vc<t> a,u x){
return a-=x;
}
//N() が単位元
//VERIFY: yosupo
//CF407E
template<class N>
struct segtree{
vc<N> x;
int n,s;
segtree(){}
template<class t>
segtree(const vc<t>&a){
n=a.size();
s=1;
while(s<n){s*=2;}
x.resize(s*2);
rep(i,n)
x[s+i]=N(a[i]);
gnr(i,1,s)
x[i]=N::merge(x[i*2],x[i*2+1]);
}
//NOT Verified
segtree(int nn){
resize(nn);
}
void resize(int nn){
n=nn;
s=1;
while(s<n){s*=2;}
x.assign(s*2,N());
gnr(i,1,s)
x[i]=N::merge(x[i*2],x[i*2+1]);
}
template<class t>
void load(const vc<t>&a){
n=a.size();
s=1;
while(s<n){s*=2;}
x.resize(s*2);
rep(i,n)
x[s+i]=N(a[i]);
rng(i,n,s)
x[s+i]=N();
gnr(i,1,s)
x[i]=N::merge(x[i*2],x[i*2+1]);
}
void clear(){
rep(i,n)
x[s+i]=N();
gnr(i,1,s)
x[i]=N::merge(x[i*2],x[i*2+1]);
}
N point_get(int i){
assert(inc(0,i,n-1));
return x[i+s];
}
void point_set(int i,const N&t){
assert(inc(0,i,n-1));
i+=s;
x[i]=t;
while(i>>=1)x[i]=N::merge(x[i*2],x[i*2+1]);
}
void point_merge(int i,const N&t){
assert(inc(0,i,n-1));
i+=s;
x[i]=N::merge(x[i],t);
while(i>>=1)x[i]=N::merge(x[i*2],x[i*2+1]);
}
template<class F,class...Args>
void point_change(int i,F f,Args&&...args){
assert(inc(0,i,n-1));
i+=s;
(x[i].*f)(forward<Args>(args)...);
while(i>>=1)x[i]=N::merge(x[i*2],x[i*2+1]);
}
N composite(int b,int e){
assert(0<=b&&b<=e&&e<=n);
N lf,rt;
for(int l=b+s,r=e+s;l<r;l>>=1,r>>=1){
if (l&1){
lf=N::merge(lf,x[l]);
l++;
}
if (r&1){
r--;
rt=N::merge(x[r],rt);
}
}
return N::merge(lf,rt);
}
N getall(){
return x[1];
}
//UTPC2020E
//n 超えるかもしれない
template <class F,class... Args>
pair<int,N> max_right(int l,F f,Args&&... args){
assert((N().*f)(forward<Args>(args)...));
assert(0<=l&&l<=n);
if(l==n)return mp(n,N());
l+=s;
N sm;
assert((sm.*f)(forward<Args>(args)...));
do {
while (l % 2 == 0) l >>= 1;
if (!(N::merge(sm,x[l]).*f)(forward<Args>(args)...)){
while (l < s) {
l = (2 * l);
N tmp=N::merge(sm,x[l]);
if ((tmp.*f)(forward<Args>(args)...)) {
sm = tmp;
l++;
}
}
return mp(l - s,sm);
}
sm = N::merge(sm, x[l]);
l++;
} while ((l & -l) != l);
return mp(n,sm);
}
//UTPC2020E
template <class F,class... Args>
pair<int,N> min_left(int r,F f,Args&&... args){
assert((N().*f)(forward<Args>(args)...));
assert(0<=r&&r<=n);
if(r==0)return mp(0,N());
r+=s;
N sm;
do {
r--;
while (r > 1 && (r % 2)) r >>= 1;
if (!(N::merge(x[r],sm).*f)(forward<Args>(args)...)) {
while (r < s) {
r = (2 * r + 1);
N tmp=N::merge(x[r],sm);
if ((tmp.*f)(forward<Args>(args)...)) {
sm = tmp;
r--;
}
}
return mp(r + 1 - s,sm);
}
sm = N::merge(x[r], sm);
} while ((r & -r) != r);
return mp(0,sm);
}
//行列とか乗せて必要なのはベクトルとの積，みたいなときに使える？
//CF Goodbye 2016 E
template<class F,class T>
T accumulate(int l, int r,F f,T t) {
assert(0<=l&&l<=r&&r<=s);
if (l == r) return t;
l+=s;
r+=s;
static int buf[2][30];
int cnt[2]{};
{
int l2 = l, r2 = r;
while (l < r) {
if (l & 1){
//(x[l++].*f)(forward<Args>(args)...);
buf[0][cnt[0]++]=l++;
}
if (r & 1){
//(x[--r].*f)(forward<Args>(args)...);
buf[1][cnt[1]++]=--r;
}
l >>= 1;
r >>= 1;
}
l = l2;
r = r2;
}
rep(i,cnt[0])t=(x[buf[0][i]].*f)(t);
per(i,cnt[1])t=(x[buf[1][i]].*f)(t);
return t;
}
};
//最も基本的な動的 segtree
//ByteCamp 2022 Day4 I
template<class N>
struct psegtree{
struct N2{
N v;
int l,r;
};
vc<N2> x;
int L,s;
template<class t>
psegtree(const vc<t>&a,int foresight=0):x(foresight){
x.clear();
int n=a.size();
L=0;
while((1<<L)<n)L++;
s=1<<L;
x.resize(s*2,{N(),0,0});
rep(i,n)x[s+i].v=N(a[i]);
gnr(i,1,s){
x[i].v=N::merge(x[i*2].v,x[i*2+1].v);
x[i].l=i*2;
x[i].r=i*2+1;
}
}
psegtree(int foresight=0):x(foresight){}
int init(int n){
x.clear();
L=0;
while((1<<L)<n)L++;
s=1<<L;
int a=nl(N());
rep(i,L)a=nn(a,a);
return a;
}
int nl(const N&val){
x.pb({val,-1,-1});
return si(x)-1;
}
int nn(int l,int r){
x.pb({N::merge(x[l].v,x[r].v),l,r});
return si(x)-1;
}
int point_set(int root,int i,const N&val){
static int buf[40];
rep(lv,L){
buf[lv]=root;
if((i>>(L-1-lv))&1)root=x[root].r;
else root=x[root].l;
}
int res=nl(val);
per(lv,L){
int j=buf[lv];
if((i>>(L-1-lv))&1){
res=nn(x[j].l,res);
}else{
res=nn(res,x[j].r);
}
}
return res;
}
int point_merge(int root,int i,const N&val){
static int buf[40];
rep(lv,L){
buf[lv]=root;
if((i>>(L-1-lv))&1)root=x[root].r;
else root=x[root].l;
}
int res=nl(N::merge(x[root].v,val));
per(lv,L){
int j=buf[lv];
if((i>>(L-1-lv))&1){
res=nn(x[j].l,res);
}else{
res=nn(res,x[j].r);
}
}
return res;
}
N compositedfs(int root,int l,int r,int b,int e){
if(e<=l||r<=b)return N();
if(b<=l&&r<=e)return x[root].v;
int mid=(l+r)/2;
return N::merge(compositedfs(x[root].l,l,mid,b,e),compositedfs(x[root].r,mid,r,b,e));
}
N composite(int root,int b,int e){
assert(b<=e);
if(b==e)return N();
return compositedfs(root,0,s,b,e);
}
};
struct MaxNode{
int v;
MaxNode(int vv=-inf):v(vv){}
static MaxNode merge(const MaxNode&a,const MaxNode&b){
return MaxNode(max(a.v,b.v));
}
bool ok(int x){return v<x;}
};
struct MinNode{
pi v;
MinNode(pi vv=pi(inf,-1)):v(vv){}
static MinNode merge(const MinNode&a,const MinNode&b){
return MinNode(min(a.v,b.v));
}
};
struct SumNode{
int v;
SumNode(int vv=0):v(vv){}
static SumNode merge(const SumNode&a,const SumNode&b){
return SumNode(a.v+b.v);
}
};
segtree<MinNode> minseg;
psegtree<SumNode> z(ten(5)*31*2*2+12345);
vi ls[2];
void init(int n,vi h){
vc<pi> hi(n);
rep(i,n)hi[i]=pi(h[i],i);
minseg.load(hi);
int root=z.init(ten(9));
rep(phase,2){
dmp(phase);
segtree<MaxNode> mx(h);
ls[phase].resize(n);
int cur=root;
auto add=[&](int i,int v){
if(i<0)return;
dmp2(i,v);
cur=z.point_merge(cur,i,v);
};
vi pos;
rep(i,n){
dmp(i);
while(si(pos)&&h[pos.back()]>h[i]){
int j=gpp(pos);
dmp2(i,j);
int v=mx.composite(j+1,i).v;
if(si(pos)){
int k=pos.back();
int w=mx.composite(k+1,j).v;
add(w-h[j],-1);
chmin(v,w);
}
add(v-h[j],1);
}
if(si(pos)){
int k=pos.back();
int w=mx.composite(k+1,i).v;
add(w-h[i],1);
}
pos.pb(i);
ls[phase][i]=cur;
}
reverse(all(h));
}
reverse(all(ls[1]));
}
int max_towers(int l,int r,int d){
int m=minseg.composite(l,r+1).v.b;
dmp2(l,m,r);
int ans=0;
ans+=z.composite(ls[0][r],d,z.s).v;
ans-=z.composite(ls[0][m],d,z.s).v;
dmp(ans);
ans+=z.composite(ls[1][l],d,z.s).v;
ans-=z.composite(ls[1][m],d,z.s).v;
dmp(ans);
ans++;
dmp(ans);
return ans;
}
