#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
#include "gondola.h"
using namespace std;
using namespace __gnu_pbds;
typedef tree<int, null_type, less<int>, rb_tree_tag,tree_order_statistics_node_update> ordered_set;
#define scan(x) do{while((x=getchar())<'0'); for(x-='0'; '0'<=(_=getchar()); x=(x<<3)+(x<<1)+_-'0');}while(0)
char _;
#define complete_unique(a) a.erase(unique(a.begin(),a.end()),a.end())
#define all(a) a.begin(),a.end()
#define println printf("\n");
#define readln(x) getline(cin,x);
#define pb push_back
#define endl "\n"
#define INT_INF 0x3f3f3f3f
#define LL_INF 0x3f3f3f3f3f3f3f3f
#define MOD 1000000009
#define mp make_pair
#define fastio cin.tie(0); cin.sync_with_stdio(0);
#define MAXN 100001
#define MAXV 250001
typedef unsigned long long ull;
typedef long long ll;
typedef long double ld;
typedef unordered_map<int,int> umii;
typedef pair<int,int> pii;
typedef pair<double,double> pdd;
typedef pair<ll,ll> pll;
typedef pair<int,pii> triple;
typedef int8_t byte;
mt19937 g1(time(0));
int randint(int a, int b){return uniform_int_distribution<int>(a, b)(g1);}
ll randlong(ll a,ll b){return uniform_int_distribution<long long>(a, b)(g1);}
ll gcd(ll a, ll b){return b == 0 ? a : gcd(b, a % b);}
ll lcm(ll a, ll b){return a*b/gcd(a,b);}
ll fpow(ll b, ll exp, ll mod){if(exp == 0) return 1;ll t = fpow(b,exp/2,mod);if(exp&1) return t*t%mod*b%mod;return t*t%mod;}
ll divmod(ll i, ll j, ll mod){i%=mod,j%=mod;return i*fpow(j,mod-2,mod)%mod;}
int arr[MAXN] = {3,4};
set<int> s;
vector<int> getSequence(int n, int seq[MAXN]){
int idx = -1;
for(int i=0; i<n; i++){
if(seq[i] <= n){
idx = i;
break;
}
}
if(idx == -1){
vector<int> v(n); for(int i=1; i<=n; i++) v[i-1] = i;
return v;
}
for(int i=1; i<n; i++){
if(seq[(idx+i)%n] > n) continue;
int req = (seq[idx]+i)%n;
if(req == 0) req = n;
if(req != seq[(idx+i)%n]) return {-1};
}
for(int i=1; i<n; i++){
int req = (seq[idx]+i)%n;
if(req == 0) req = n;
seq[(idx+i)%n] = req;
}
vector<int> v; for(int i=0; i<n; i++) v.pb(seq[i]);
return v;
}
int valid(int n, int seq[MAXN]){
for(int i=0; i<n; i++){
if(s.count(seq[i])) return 0;
s.insert(seq[i]);
}
s.clear();
vector<int> v = getSequence(n,seq);
if(v[0] == -1) return 0;
int idx = -1;
for(int i=0; i<n; i++)
if(v[i] == 1)
idx = i;
rotate(v.begin(),v.begin()+idx,v.begin()+n);
for(int i=1; i<n; i++)
if(v[i] != v[i-1]+1)
return 0;
return 1;
}
int replacement(int n, int gondolaSeq[MAXN], int replacementSeq[MAXV]){
int tmp[MAXN]; memcpy(tmp,gondolaSeq,sizeof tmp);
vector<int> v = getSequence(n,tmp);
priority_queue<pii,vector<pii>,greater<pii>> q;
for(int i=0; i<n; i++)
q.push(mp(gondolaSeq[i],i));
int cnt = 0, mx = n;
while(q.size()){
int vv = q.top().first, idx = q.top().second; q.pop();
while(v[idx] < vv){
replacementSeq[cnt++] = v[idx];
v[idx] = ++mx;
}
}
return cnt;
}
int countReplacement(int n, int seq[MAXN]){
int a[MAXN]; memcpy(a,seq,sizeof a);
if(!valid(n,a)) return 0;
vector<int> v;
for(int i=0; i<n; i++)
if(seq[i] > n)
v.pb(seq[i]);
int rem_choose = (int)v.size(), lst = n;
ll res = (rem_choose==n?n:1);
sort(all(v));
for(int i=0; i<v.size(); i++){
int options = v[i]-lst-1;
res = ((res*fpow(rem_choose,options,MOD))%MOD+MOD)%MOD;
rem_choose--;
lst = v[i];
}
return (int)res;
}
