#include "rect.h"
#include <bits/stdc++.h>
using namespace std;
#define forn(i,n) for(int i=0; i<n; ++i)
#define pi pair<int,int>
#define f first
#define s second
#define ll long long
#define pb push_back
#define all(x) x.begin(),x.end()
struct DSU {
vector<int> p, sz, l, r, t, d;
int m;
DSU(int n, int _m) {
m=_m;
forn(i,n*m) {
p.pb(i);
sz.pb(1);
l.pb(i%m);
r.pb(i%m);
t.pb(i/m);
d.pb(i/m);
}
}
int get(int u) {
return p[u]==u?u:get(p[u]);
}
void uni(int u, int v) {
u=get(u), v=get(v);
if (u==v) return;
if (sz[u]<sz[v]) swap(u,v);
p[v]=u;
sz[u]+=sz[v];
l[u]=min(l[u],l[v]);
r[u]=max(r[u],r[v]);
t[u]=min(t[u],t[v]);
d[u]=max(d[u],d[v]);
}
void uni(int a, int b, int c, int d) {
uni(a*m+b,c*m+d);
}
};
ll wrong(vector<vector<int>> b) {
int n=b.size(), m=b[0].size();
vector<vector<int>> a(n+2,vector<int>(m+2,-1));
forn(i,n) forn(j,m) a[i+1][j+1]=b[i][j];
vector<pi> v;
n+=2, m+=2;
DSU dsu(n,m);
forn(i,n) forn(j,m) if (a[i][j]!=-1) v.pb({a[i][j],i*m+j});
sort(all(v));
vector<vector<int>> ok(n,vector<int>(m,0));
forn(i,n) forn(j,m) {
if (i==0 || i==n-1) ok[i][j]=1;
if (j==0 || j==m-1) ok[i][j]=1;
}
forn(i,n) forn(j,m) if (ok[i][j]) dsu.uni(i,j,0,0);
ll ans=0;
int N=v.size();
for(int i=0; i<N; ) {
vector<int> z;
int x=v[i].f;
while (i<N && v[i].f==x) {
z.pb(v[i].s);
++i;
}
for(auto&it:z) {
int x=it/m, y=it%m;
if (ok[x-1][y]) dsu.uni(x-1,y,x,y);
if (ok[x+1][y]) dsu.uni(x+1,y,x,y);
if (ok[x][y-1]) dsu.uni(x,y-1,x,y);
if (ok[x][y+1]) dsu.uni(x,y+1,x,y);
ok[x][y]=1;
}
map<int,int> was;
for(auto&it:z) {
int x=it/m, y=it%m;
int u=dsu.get(it);
if (was[u]) continue;
was[u]=1;
int b=dsu.l[u];
int a=dsu.t[u];
int d=dsu.r[u];
int c=dsu.d[u];
int k=(c-a+1)*(d-b+1);
if (k==dsu.sz[u]) ++ans;
}
}
return ans-1;
}
const int N=700;
bitset<N> dp1[N][N], dp2[N][N];
struct BIT {
int n,m;
vector<vector<int>> t;
BIT(int n, int m) {
this->n=n, this->m=m;
t.assign(n,vector<int>(m,0));
}
void addi(int i, int j, int x) {
for(;j<m;j=j|(j+1)) t[i][j]+=x;
}
void add(int i, int j, int x) {
for(;i<n;i=i|(i+1)) addi(i,j,x);
}
int sumi(int i, int j) {
int q=0;
for(;j>=0;j=(j&(j+1))-1) q+=t[i][j];
return q;
}
int sum(int a, int b) {
int q=0;
for(;a>=0;a=(a&(a+1))-1) q+=sumi(a,b);
return q;
}
int sum(int a, int b, int c, int d) {
return sum(c,d)-sum(a-1,d)-sum(c,b-1)+sum(a-1,b-1);
}
};
struct dsu {
vector<int> p,sz,l,r;
dsu(int n) {
forn(i,n) p.pb(i), sz.pb(1);
l=r=p;
}
int get(int u) {
return p[u]==u?u:get(p[u]);
}
void uni(int u, int v) {
u=get(u),v=get(v);
if (u==v) return;
if (sz[u]<sz[v]) swap(u,v);
p[v]=u;
sz[u]+=sz[v];
l[u]=min(l[u],l[v]);
r[u]=max(r[u],r[v]);
}
};
vector<dsu> dsu1(N,dsu(N)), dsu2(N,dsu(N));
int ok[N][N];
BIT ft(N,N);
int a[N][N];
int pre[N+1];
const int K=11;
int st1[N][N][K], st2[N][N][K];
void init() {
int p=0;
forn(i,N+1) {
while (2*(1<<p) < i) ++p;
pre[i]=p;
}
}
void build(int k, vector<int>&a) {
int n=a.size();
forn(i,n) st1[k][i][0]=a[i];
for(int j=1; j<K; ++j) {
forn(i,n) st1[k][i][j]=max(st1[k][i][j-1], (i+(1<<(j-1)))<n ? st1[k][i+(1<<(j-1))][j-1] : 0 );
}
}
int query1(int k, int l, int r) {
if (l==r) return st1[k][l][0];
int p=pre[r-l+1];
return max( st1[k][l][p] , st1[k][r-(1<<p)+1][p] );
}
int query2(int k, int l, int r) {
if (l==r) return st2[k][l][0];
int p=pre[r-l+1];
return max( st2[k][l][p] , st2[k][r-(1<<p)+1][p] );
}
int it=0;
int check(int l, int u, int r, int d) {
int ok=1;
for (int i=l; i<=r; ++i) {
int x=query1(i,u,d);
ok&=a[i][u-1]>x && x<a[i][d+1];
++it;
}
for (int i=u; i<=d; ++i) {
int x=query2(i,l,r);
ok&=a[l-1][i]>x && x<a[r+1][i];
++it;
}
if (it>2e8) exit(1);
return ok;
}
ll count_rectangles(vector<vector<int>> b) {
int n=b.size(), m=b[0].size();
if (min(n,m)<3) return 0;
int mx=0; for(auto&x:b) for(auto&y:x) mx=max(mx,y);
if (mx==1) return wrong(b);
if (n==3) {
ll ans=0;
for (int i=0; i<m; ++i) {
int mx=0;
for (int j=i+1; j<m-1; ++j) {
if (b[0][j]<=b[1][j] || b[1][j]>=b[2][j]) break;
mx=max(mx,b[1][j]);
ans += b[1][i]>mx && mx<b[1][j+1];
}
}
return ans;
}
if (max(n,m)>N) exit(0);
init();
forn(i,n) forn(j,m) a[i][j]=b[i][j];
forn(i,n) build(i,b[i]);
swap(st1,st2);
forn(j,m) {
vector<int> v; forn(i,n) v.pb(a[i][j]);
build(j,v);
}
swap(st1,st2);
vector<pi> v;
forn(i,n) forn(j,m) v.pb({a[i][j],i*m+j});
sort(all(v));
ll ans=0;
int N=v.size();
map<ll,ll> was;
for(int i=0; i<N; ) {
vector<int> z;
int x=v[i].f;
while (i<N && v[i].f==x) {
z.pb(v[i].s);
++i;
}
for(auto&it:z) {
int x=it/m, y=it%m;
if (x) if (ok[x-1][y]) dsu1[y].uni(x-1,x);
if (x<n-1) if (ok[x+1][y]) dsu1[y].uni(x+1,x);
if (y) if (ok[x][y-1]) dsu2[x].uni(y-1,y);
if (y<m-1) if (ok[x][y+1]) dsu2[x].uni(y+1,y);
ok[x][y]=1;
ft.add(x,y,1);
}
for(auto&it:z) {
int x=it/m, y=it%m;
int u=dsu1[y].get(x);
int a=dsu1[y].l[u], c=dsu1[y].r[u];
int v=dsu2[x].get(y);
int b=dsu2[x].l[v], d=dsu2[x].r[v];
if (a==0 || b==0 || c==n-1 || d==m-1) continue;
if (was[a*m+b+n*m*(c*m+d)]) continue;
was[a*m+b+n*m*(c*m+d)]=1;
if (min(n,m)<700) {
int q = check(a,b,c,d);
ans+=q;
} else {
it+=(c-a+1)+(d-b+1);
}
}
}
assert(it<=2e8);
return ans;
}
