#include "gondola.h"
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace __gnu_pbds;
#define watch(x) cout<<(#x)<<"="<<(x)<<'\n'
#define mset(d,val) memset(d,val,sizeof(d))
#define setp(x) cout<<fixed<<setprecision(x)
#define forn(i,a,b) for(int i=(a);i<(b);i++)
#define fore(i,a,b) for(int i=(a);i<=(b);i++)
#define pb push_back
#define F first
#define S second
#define pqueue priority_queue
#define fbo find_by_order
#define ook order_of_key
typedef long long ll;
typedef pair<ll,ll> ii;
typedef vector<ll> vi;
typedef vector<ii> vii;
typedef long double ld;
typedef tree<ll, null_type, less<ll>, rb_tree_tag, tree_order_statistics_node_update> pbds;
void amin(ll &a, ll b){ a=min(a,b); }
void amax(ll &a, ll b){ a=max(a,b); }
void YES(){cout<<"YES\n";} void NO(){cout<<"NO\n";}
void SD(int t=0){ cout<<"PASSED "<<t<<endl; }
const ll INF = ll(1e18);
const int MOD = 1e9+9;
bool DEBUG = 0;
const int MAXN = 100005;
const int MAXA = 250005;
ll add(ll a,ll b)
{
a+=b; a%=MOD;
if(a<0) a+=MOD;
return a;
}
ll mult(ll a, ll b)
{
if(a>MOD) a%=MOD;
if(b>MOD) b%=MOD;
ll ans=(a*b)%MOD;
if(ans<0) ans+=MOD;
return ans;
}
ll pw(ll a, ll b)
{
ll r=1;
while(b){
if(b&1) r=mult(r,a);
a=mult(a,a);
b>>=1;
}
return r;
}
ll inverse(ll a)
{
return pw(a,MOD-2);
}
int valid(int n, int inputSeq[])
{
int minid=0;
forn(i,0,n){
if(inputSeq[minid]>inputSeq[i]){
minid=i;
}
}
int tmp[n];
forn(i,0,n){
tmp[i]=inputSeq[(minid+i)%n];
}
forn(i,0,n) inputSeq[i]=tmp[i];
set<ll> used;
int last=-1;
for(int i=0;i<n;i++){
if(used.find(inputSeq[i])!=used.end()) return 0;
used.insert(inputSeq[i]);
if(inputSeq[i]<=n){
if(last!=-1 && inputSeq[i]-inputSeq[last]!=i-last) return 0;
last=i;
}
}
return 1;
}
int replacement(int n, int gondolaSeq[], int replacementSeq[])
{
assert(valid(n, gondolaSeq));
int loc[MAXA];
mset(loc,-1);
int large=0;
for(int i=0;i<n;i++){
loc[gondolaSeq[i]]=i;
if(gondolaSeq[i]>gondolaSeq[large]){
large=i;
}
}
int a[n];
int start=0,val=1;
for(int i=0;i<n;i++){
if(gondolaSeq[i]<=n){
start=i;
val=gondolaSeq[i];
break;
}
}
for(int i=start;;){
a[i]=val++;
if(val==n+1) val=1;
i=(i+1)%n;
if(i==start) break;
}
int ptr=0;
for(int i=n+1;i<=gondolaSeq[large];i++){
if(loc[i] == -1){
//cout<<"Not found\n";
replacementSeq[ptr++]=a[large];
a[large]=i;
}
else{
//cout<<"Found\n";
replacementSeq[ptr++] = a[loc[i]];
a[loc[i]]=i;
}
}
return ptr;
}
int countReplacement(int n, int inputSeq[])
{
if(!valid(n, inputSeq)) return 0;
vi v;
for(int i=0;i<n;i++){
v.pb(inputSeq[i]);
}
sort(v.begin(), v.end(), greater<ll>());
ll ans=1;
if(v.back()>n) ans=n;
ll cur=0;
while(v.size() && v.back()<=n) v.pop_back();
cur=v.size();
v.pb(n);
reverse(v.begin(), v.end());
for(int i=1;i<v.size();i++){
ans=mult(ans, pw(cur, v[i]-v[i-1]-1));
cur--;
}
return ans;
}
