#pragma GCC optimize("Ofast")
#include "gondola.h"
#include<bits/stdc++.h>
using namespace std;
#define pb push_back
#define fi first
#define se second
#define sz(a) (int)a.size()
#define all(a) a.begin(),a.end()
#define lb lower_bound
#define ub upper_bound
#define owo ios_base::sync_with_stdio(0);cin.tie(0);
#define MOD (ll)(1e9+9)
#define INF (ll)(1e18)
#define debug(...) fprintf(stderr, __VA_ARGS__),fflush(stderr)
#define time__(d) for(long blockTime = 0; (blockTime == 0 ? (blockTime=clock()) != 0 : false);\
debug("%s time : %.4fs\n", d, (double)(clock() - blockTime) / CLOCKS_PER_SEC))
typedef long long int ll;
typedef long double ld;
typedef pair<ll,ll> PII;
typedef pair<int,int> pii;
typedef vector<vector<int>> vii;
typedef vector<vector<ll>> VII;
ll gcd(ll a,ll b){if(!b)return a;else return gcd(b,a%b);}
int valid(int n, int inputSeq[])
{
vector<int>a(n);
for(int i=0;i<n;i++)a[i] = inputSeq[i];
int mn=1e9,id=0,mx = 0;
bool ok = 1;
map<int,bool>seen;
vector<pii>pos;
for(int i=0;i<n;i++){
if(a[i] < mn){
mn = a[i];
id = i;
}
if(seen[a[i]])ok = 0;
seen[a[i]] = 1;
mx = max(mx,a[i]);
}
vector<int>b;
for(int i=0;i<n;i++){
b.pb(a[(id+i)%n]);
}
for(int i=0;i<n;i++){
if(b[i]<=n)pos.pb({b[i],i});
}
// for(auto x:pos)cout<<x.fi<<" "<<x.se<<'\n';
if(mx == n){
sort(all(a));
if(a != b)ok = 0;
}else{
mx = 0;
for(int i=0;i<sz(pos);i++){
if(i && pos[i].se - pos[i-1].se != pos[i].fi - pos[i-1].fi)ok = 0;
}
}
return ok;
}
int replacement(int n, int gondolaSeq[], int replacementSeq[])
{
vector<int>a(n);
for(int i=0;i<n;i++)a[i] = gondolaSeq[i];
map<int,bool>seen;
int mx = 0;
for(int i=0;i<n;i++){
seen[a[i]] = 1;
mx = max(mx,a[i]);
}
int ans = 0;
bool fix = 0;
for(int i=1;i<=n;i++)if(seen[i])fix = 1;
if(fix){
vector<int>b(n);
int sh = 0;
for(int i=0;i<n;i++){
if(a[i] <= n){
sh = i - (a[i]-1);
break;
}
}
for(int i=0;i<n;i++)b[i] = a[(i+sh+n)%n];
for(int i=0;i<n;i++)a[i] = b[i];
}
vector<pii>pos;
for(int i=0;i<n;i++){
if(a[i] > n)pos.pb({a[i],i+1});
}
sort(all(pos),greater<pii>());
vector<int>opt;
for(int i=0;i<sz(pos);i++){
int num;
if(i != sz(pos)-1)num = pos[i].fi-pos[i+1].fi;
else num = pos[i].fi - n;
ans+=num;
while(num--){
opt.pb(pos[i].se);
}
}
reverse(all(opt));
vector<int>c(n+1);
for(int i=1;i<=n;i++)c[i] = i;
int cnt = n;
int i = 0;
for(int x:opt){
replacementSeq[i] = c[x];
i++;
c[x] = ++cnt;
}
return ans;
}
int countReplacement(int n, int inputSeq[])
{
if(!valid(n,inputSeq))return 0;
vector<int>a(n);
for(int i=0;i<n;i++)a[i] = inputSeq[i];
map<int,bool>seen;
int mx = 0;
for(int i=0;i<n;i++){
seen[a[i]] = 1;
mx = max(mx,a[i]);
}
bool fix = 0;
for(int i=1;i<=n;i++)if(seen[i])fix = 1;
if(fix){
vector<int>b(n);
int sh = 0;
for(int i=0;i<n;i++){
if(a[i] <= n){
sh = i - (a[i]-1);
break;
}
}
for(int i=0;i<n;i++)b[i] = a[(i+sh+n)%n];
for(int i=0;i<n;i++)a[i] = b[i];
}
vector<pii>pos;
for(int i=0;i<n;i++){
if(a[i] > n)pos.pb({a[i],i+1});
}
sort(all(pos),greater<pii>());
vector<int>opt;
for(int i=0;i<sz(pos);i++){
int num;
if(i != sz(pos)-1)num = pos[i].fi-pos[i+1].fi;
else num = pos[i].fi - n;
while(num--){
opt.pb(pos[i].se);
}
}
map<int,bool>diff;
for(int x:opt)diff[x] = 1;
int s = sz(diff); //there are only s fix points in opt
//the rest we can just go wild;
int ans = 1;
for(int i=0;i<sz(opt)-s;i++){
ans = (ans*s)%MOD;
}
if(!fix)ans = (ans*n)%MOD;
return ans;
}
