#ifndef LOCAL
#pragma GCC optimize ("Ofast")
#pragma GCC optimize ("unroll-loops")
#endif
#include "aliens.h"
#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<<"}";
}
template<class t>
void print(t x,int suc=1){
cout<<x;
if(suc==1)
cout<<"\n";
if(suc==2)
cout<<" ";
}
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,class u>
void print(const pair<t,u>&p,int suc=1){
print(p.a,2);
print(p.b,suc);
}
template<class t,class u>
void print_offset(const pair<t,u>&p,ll off,int suc=1){
print(p.a+off,2);
print(p.b+off,suc);
}
template<class T>
void print(const vector<T>&v,int suc=1){
rep(i,v.size())
print(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(v[i]+off,i==int(v.size())-1?suc:2);
}
template<class T,size_t N>
void print(const array<T,N>&v,int suc=1){
rep(i,N)
print(v[i],i==int(N)-1?suc:2);
}
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);
}
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);
}
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>
void transvvc(int&n,int&m,vvc<t>&a){
assert(si(a)==n);
vvc<int> b(m,vi(n));
rep(i,n){
assert(si(a[i])==m);
rep(j,m)b[j][i]=a[i][j];
}
a.swap(b);
swap(n,m);
}
//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>
t MIN(const vc<t>&a){
return *min_element(all(a));
}
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);
}
//cost(i,j) が monge
//O(N) で DP する
//stress-tested
template<class F>
auto&larsch(int n,F cost){
assert(2<=n);
using t=typename invoke_result<F,int,int>::type;
const int L=topbit(n);
static vc<t> res;res.resize(n);
static vc<pair<t,int>> val[30];
static vc<pi> pos[30];
static int s[30];
rep(i,L){
val[i].resize(n>>i);
pos[i].resize(n>>i);
s[i]=0;
}
s[0]=n;
rep(i,n)pos[0][i].a=i;
auto eval=[&](int lv,int i,int j){
return res[pos[lv][i].a]+cost(pos[lv][i].a,((j+1)<<lv)-1);
};
auto setval=[&](int lv,int i,int j){
val[lv][j]=mp(eval(lv,i,j),i);
};
res[0]=t{};
rng(i,1,n){
const int l=i,r=i+1;
int lv=botbit(l);
if((l>>lv)<(n>>lv)){
setval(lv,(l>>lv)-1,l>>lv);
if(l>>(lv+1))
chmin(val[lv][l>>lv],
mp(val[lv+1][l>>(lv+1)].a,pos[lv+1][val[lv+1][l>>(lv+1)].b].b));
}
while(lv>0){
lv--;
if((l>>lv)<(n>>lv)){
int lw=val[lv][(l>>lv)-1].b;
int up=(l>>(lv+1))<(n>>(lv+1))?
pos[lv+1][val[lv+1][l>>(lv+1)].b].b:(l>>lv)-1;
assert(lw<=up);
setval(lv,lw,l>>lv);
rng(j,lw+1,up+1)
chmin(val[lv][l>>lv],mp(eval(lv,j,l>>lv),j));
}
}
res[i]=val[0][i].a;
while((r&1<<lv)==0&&lv<L-1){
lv++;
rng(b,(r>>(lv-1))-2,r>>(lv-1)){
int a=pos[lv-1][b].a;
while(s[lv]>=(r>>lv)){
if(eval(lv,s[lv]-1,s[lv])>res[a]+cost(a,((s[lv]+1)<<lv)-1))s[lv]--;
else break;
}
if(s[lv]+1<(n>>lv))pos[lv][s[lv]++]=pi(a,b);
}
}
}
return res;
}
//0->n-1 のジャンプ
//monge コストでちょうど k step でやれ，みたいな最小化問題
//ans(step) が単調減少だとする(つまり小刻みであればあるほど嬉しい)
//max(ans(step))-max(ans(step)) の上界を dif で与えている
//stress-tested (define ll なし，復元もちゃんと確かめた)
template<class F>
ll kstepmin(int n,int k,ll dif,F cost,vi*sol=nullptr){
assert(inc(1,k,n-1));
ll lw=-1,up=dif+1,ans=infLL;
vc<pair<ll,int>> buf(n);
while(up-lw>1){
ll mid=(lw+up)/2;
auto&dp=larsch(n,[&](int i,int j){
return pair<ll,int>(cost(i,j)+mid,1);
});
if(dp[n-1].b<=k){
up=mid;
ans=dp[n-1].a-k*mid;
copy(all(dp),buf.bg);
}else lw=mid;
}
assert(ans<infLL);
if(sol){
sol->clear();
int cur=n-1;
sol->pb(cur);
while(cur){
assert(k);
k--;
int nx=cur;
while(nx--){
if(buf[nx].b<=k&&buf[nx].a+cost(nx,cur)+up==buf[cur].a){
sol->pb(cur=nx);
break;
}
}
assert(nx>=0);
}
assert(k==0);
reverse(sol->bg,sol->ed);
}
return ans;
}
long long take_photos(int n, int m, int k, std::vector<int> r, std::vector<int> c){
vc<pi> rc(n);
rep(i,n)rc[i]=minmax(r[i],c[i]);
sort(all(rc),[&](pi a,pi b){
return a.b!=b.b?a.b<b.b:a.a>b.a;
});
int s=0;
for(auto [i,j]:rc){
while(s&&rc[s-1].a>=i)s--;
rc[s++]=pi(i,j);
}
rc.resize(n=s);
auto f=[&](int i,int j){
assert(0<=i&&i<j&&j<=n);
ll res=sq<ll>(rc[j-1].b+1-rc[i].a);
if(i>0&&rc[i].a<rc[i-1].b+1){
res-=sq<ll>(rc[i-1].b+1-rc[i].a);
}
return res;
};
return kstepmin(n+1,min(k,n),sq<ll>(m),f);
}
