#include <bits/stdc++.h>
#define jizz ios::sync_with_stdio(0),cin.tie(0),cout.tie(0);
#define pb push_back
#define ET cout << "\n"
#define MEM(i,j) memset(i,j,sizeof i)
#define F first
#define S second
#define MP make_pair
#define ALL(v) v.begin(),v.end()
#define DB(a,s,e) {for(int i=s;i<e;++i) cout << a[i] << " ";ET;}
using namespace std;
typedef long long ll;
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;
const ll MOD=1e9;
ll dx[4]={1,-1,0,0},dy[4]={0,0,1,-1},ans,boss[100005];
pll val[100005],dots[100005];
vector<ll> v,hav[100005];
unordered_map<ll,ll> mpp;
vector<ll> G[100005];
bitset<100005> vis;
ll finds(ll x)
{
if(x==boss[x]) return x;
return boss[x]=finds(boss[x]);
}
void Union(ll a,ll b)
{
a=finds(a),b=finds(b);
if(a==b) return;
boss[a]=b;
}
bool yee(ll a,ll b)
{
return dots[a].S>dots[b].S;
}
void dfs(int u,int f)
{
vis[u]=1;
sort(ALL(hav[u]),yee);
ll L=hav[u].size(),U=0,D=hav[u].size()-1,F,S,X=0,Y=0;
ll ucat=-1,dcat=-1;
map<ll,pll> mp;
ans=(ans+(L-1)*L*(L+1)/6)%MOD;
for(ll i:hav[u])
{
Y=(Y+X)%MOD;
for(ll j:G[i])
if(boss[j]!=f)
{
if(!vis[boss[j]]) dfs(boss[j],u);
F=val[j].F,S=val[j].S;
ans=(ans+S*U+((U+1)*(U+2)/2-1)*F+F+S+S*D+((D+1)*(D+2)/2-1)*F)%MOD;
val[i]=(MP(val[i].F+F,(val[i].S+F+S)%MOD));
auto p=mp.find(boss[j]);
if(p!=mp.end())
{
p->S.S+=p->S.F;
X-=p->S.F,Y=(Y-p->S.S+MOD)%MOD;
ans=(ans+F*Y+X*S+X*F)%MOD;
X+=p->S.F,Y=(Y+p->S.S)%MOD;
p->S.F+=F,p->S.S=(p->S.S+F+S)%MOD;
}
else
{
ans=(ans+F*Y+X*S+X*F)%MOD;
mp[boss[j]]=MP(F,F+S);
}
X+=F,Y=(Y+F+S)%MOD;
}
else
{
if(!~ucat) ucat=i;
dcat=i;
}
++U,--D;
}
X=0,Y=0;
for(ll i:hav[u])
if(i!=ucat)
Y=(Y+X)%MOD,X+=val[i].F,Y=(Y+val[i].S)%MOD;
else
{
Y=(Y+X)%MOD,val[i].F+=X,val[i].S=(val[i].S+Y)%MOD;
break;
}
reverse(ALL(hav[u]));
X=0,Y=0;
for(ll i:hav[u])
if(i!=dcat)
Y=(Y+X)%MOD,X+=val[i].F,Y=(Y+val[i].S)%MOD;
else
{
Y=(Y+X)%MOD,val[i].F+=X,val[i].S=(val[i].S+Y)%MOD;
break;
}
}
int DistanceSum(int N, int *X, int *Y)
{
for(int i=0;i<N;++i)
dots[i]=MP(Y[i],X[i]),boss[i]=i,mpp[(ll)Y[i]<<31|X[i]]=i;
for(int i=0;i<N;++i)
for(int j=2;j<4;++j)
{
auto p=mpp.find(((ll)(Y[i]+dx[j])<<31)|(X[i]+dy[j]));
if(p!=mpp.end()) Union(i,p->S);
}
for(int i=0;i<N;++i)
hav[finds(i)].pb(i),val[i]=MP(1,0);
for(int i=0;i<N;++i)
for(int j=0;j<2;++j)
{
auto p=mpp.find(((ll)(Y[i]+dx[j])<<31)|(X[i]+dy[j]));
if(p!=mpp.end()) G[i].pb(p->S);
}
dfs(finds(0),-1);
return ans;
}
