#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
const int mod=1000000000;
int add(int a,int b){return a+b-(a+b>=mod?mod:0);}
int mul(int a,int b){return (ll(a)*b)%mod;}
int res=0;
const int N=100000;
int n;
vector<array<int,2>> tree[4*N];
vector<vector<array<int,2>>> upd(4*N);
int p[N];
int sz[N];
int events=0;
int find_set(int a)
{
if(a==p[a]) return a;
else return find_set(p[a]);
}
//~ void dsu_state()
//~ {
//~ cout << "DSU state: ";
//~ map<int,vector<int>> m;
//~ for(int i=0;i<n;i++) m[find_set(i)].push_back(i);
//~ cout << "groups(" << m.size() << "):" << endl;
//~ for(auto [u,v]:m)
//~ {
//~ for(int x:v) cout << x << " ";
//~ cout << "actual_size=" << v.size() << ", sz=" << sz[u] << endl;
//~ }
//~ cout << endl;
//~ }
void merge_sets(int a,int b,vector<array<int,2>> &u)
{
a=find_set(a);
b=find_set(b);
if(a==b) return;
if(sz[a]<sz[b]) swap(a,b);
//~ cout << "about to merge [" << a << "," << b << "], before:";
//~ dsu_state();
u.push_back({b,p[b]});
u.push_back({a,sz[a]});
p[b]=a;
sz[a]+=sz[b];
//~ cout << "after:";
//~ dsu_state();
}
void apply(int idx,int l,int r,int ql,int qr,int a,int b)
{
if(ql>qr) return;
if(l==ql&&r==qr) tree[idx].push_back({a,b});
else
{
int m=(l+r)/2;
apply(2*idx,l,m,ql,min(qr,m),a,b);
apply(2*idx+1,m+1,r,max(ql,m+1),qr,a,b);
}
}
void process(int idx,int l,int r)
{
//~ cout << "idx merges[" << idx << "]: l=" << l << ", r=" << r << ": ";
for(auto [a,b]:tree[idx])
{
merge_sets(a,b,upd[idx]);
//~ cout << "[" << a << "," << b << "] ";
}
//~ cout << endl;
//~ if(l==r)
//~ {
//~ cout << "after entering and processing tree[" << l << "," << r << "]: ";
//~ dsu_state();
//~ }
if(l<r)
{
int m=(l+r)/2;
process(2*idx,l,m);
process(2*idx+1,m+1,r);
}
else
{
//~ cout << "event #" << l << ": " << sz[find_set(0)] << endl;
res=add(res,mul(sz[find_set(0)],n-sz[find_set(0)]));
}
reverse(upd[idx].begin(),upd[idx].end());
for(int i=0;i<(int)upd[idx].size();i++) ((i&1)?p:sz)[upd[idx][i][0]]=upd[idx][i][1];
}
void go(int *X,int *Y)
{
vector<array<int,2>> blocks(n);
for(int i=0;i<n;i++) blocks[i]={X[i],Y[i]};
for(int i=0;i<n;i++)
{
p[i]=i;
sz[i]=1;
}
events=0;
map<array<int,2>,int> m;
for(int i=0;i<n;i++) m[{X[i],Y[i]}]=i;
auto id=[&](int x,int y)
{
if(m.find({x,y})!=m.end()) return m[{x,y}];
else return -1;
};
vector<array<int,4>> v;
for(int i=0;i<n;i++)
{
int t=id(X[i]+1,Y[i]);
if(t!=-1) v.push_back({X[i],Y[i],i,t});
int j=id(X[i],Y[i]+1);
if(j!=-1) merge_sets(i,j,upd[0]);
}
sort(v.begin(),v.end());
int len=v.size();
vector<array<int,2>> groups;
int l=0;
while(l<len)
{
int r=l;
while(r+1<len&&v[r+1][0]==v[l][0]&&v[r][1]+1==v[r+1][1]) r++;
groups.push_back({l,r});
events++;
l=r+1;
}
for(int i=1;i<=events;i++)
{
//~ cout << "group " << i << ": ";
auto [one,two]=groups[i-1];
for(int j=one;j<=two;j++)
{
//~ cout << "[" << v[j][2] << "," << v[j][3] << "] ";
if(i-1>=1) apply(1,1,events,1,i-1,v[j][2],v[j][3]);
if(i+1<=events) apply(1,1,events,i+1,events,v[j][2],v[j][3]);
}
//~ cout << endl;
}
process(1,1,events);
//spring cleaning
for(int i=0;i<4*N;i++)
{
tree[i].clear();
upd[i].clear();
}
}
int DistanceSum(int _n,int *x,int *y)
{
n=_n;
go(x,y);
go(y,x);
return res;
}
