#include <iostream>
#include <algorithm>
#include <vector>
#include <random>
#include <chrono>
#include <set>
#include <map>
#include <stack>
#include <functional>
#include <iomanip>
#include <queue>
#include <cassert>
#include <complex>
#include <cstring>
#include <memory>
#include <bitset>
#include <sstream>
#include <cmath>
#include <numeric>
#include <numbers>
#include <fstream>
#include "gondola.h"
using namespace std;
#ifndef template
#ifndef define
#define ll long long
#define ld long double
#define pl pair<ll, ll>
#define pi pair<int, int>
#define nl cout << '\n';
#define x first
#define y second
#define cbit(x) __builtin_popcountll(x)
#define uid(a, b) uniform_int_distribution<ll>(a, b)(rng)
#define siz(x) (int)x.size()
#endif
#ifndef print
void print(size_t x) {cout << x << ' ';}
void print(int x) {cout << x << ' ';}
void print(long long x) {cout << x << ' ';}
void print(float x) {cout << x << ' ';}
void print(long double x) {cout << x << ' ';}
void print(char x) {cout << x << ' ';}
void print(const char* x) {cout << x << ' ';}
void print(bool x) {cout << x << ' ';}
void print(string &x) {cout << x << ' ';}
template<typename T, typename V>
void print(pair<T, V> &p) {print(p.x); print(p.y);}
template<typename T>
void print(vector<T> v) {for (int i = 0; i < v.size(); i++) print(v[i]);}
template<typename T>
void print(vector<vector<T>> v) {
for (int i = 0; i < v.size(); i++){
for (int j = 0; j < v[i].size(); j++)
print(v[i][j]);
nl;
}
}
template <typename T, typename... V>
void print(T t, V&&... v) {print(t); print(v...);}
#endif
#ifndef read
void read(int &x) {cin >> x;}
void read(long long &x) {cin >> x;}
void read(unsigned &x) {cin >> x;}
void read(unsigned long long &x) {cin >> x;}
void read(float &x) {cin >> x;}
void read(long double &x) {cin >> x;}
void read(char &x) {cin >> x;}
void read(string &x) {cin >> x;}
void read(bool &x) {cin >> x;}
template<typename T>
void read(vector<T> &v) {
for (int i = 0; i < v.size(); i++)
read(v[i]);
}
template <typename T, typename... V>
void read(T &t, V&... v) {read(t); read(v...);}
#endif
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
template<class T> bool maxi(T& a, const T& b) {
return a < b ? a = b, 1 : 0;
}
template<class T> bool mini(T& a, const T& b) {
return a > b ? a = b, 1 : 0;
}
template<class... Args>
auto vec(size_t n, Args&&... args) {
if constexpr(sizeof...(args) == 1)
return vector(n, args...);
else
return vector(n, vec(args...));
}
#endif
using namespace std;
const ll inf = 1e18;
const ll def = 1e6+1;
const ll mod = 1e9+9;
using namespace std;
ll powmod(ll a, ll b){
if (b == 0)
return 1;
if (b % 2 == 0){
ll val = powmod(a, b / 2);
return (val * val) % mod;
}
else
return (powmod(a, b - 1) * a) % mod;
}
int valid(int n, int inputSeq[]){
int anchor = -1;
bool res = 1;
map<int, int> mp;
for (int i = 0; i < n; i++){
mp[inputSeq[i]]++;
if (mp[inputSeq[i]] > 1)
return 0;
if (inputSeq[i] <= n){
int val = (i - inputSeq[i] + 1 + n) % n;
if (anchor == -1)
anchor = val;
else{
if (anchor != val)
res = 0;
}
}
}
return res;
}
//----------------------
int replacement(int n, int gondolaSeq[], int replacementSeq[]){
vector<int> pos;
int anchor = 0;
map<int, int> need;
for (int i = 0; i < n; i++){
if (gondolaSeq[i] <= n){
int val = (i - gondolaSeq[i] + 1 + n) % n;
anchor = val;
}
else
need[gondolaSeq[i]] = i;
}
auto org = vec(n, 0);
int j = anchor;
for (int i = 0; i < n; i++){
org[j] = i + 1;
(j += 1) %= n;
}
set<int> s;
for (int i = 0; i < n; i++){
if (org[i] != gondolaSeq[i])
s.insert(i);
}
int crr = n + 1;
while (s.size()){
int nxt;
if (need.count(crr))
nxt = need[crr];
else
nxt = *s.begin();
replacementSeq[crr - n - 1] = org[nxt];
org[nxt] = crr;
s.erase(nxt);
if (org[nxt] != gondolaSeq[nxt])
s.insert(nxt);
crr++;
}
return crr - n - 1;
}
//----------------------
int countReplacement(int n, int inputSeq[]){
if (!valid(n, inputSeq))
return 0;
vector<int> order;
for (int i = 0; i < n; i++){
if (inputSeq[i] > n)
order.push_back(inputSeq[i]);
}
sort(order.begin(), order.end());
ll res = 1, last = n + 1;
for (int i = 0; i < order.size(); i++){
(res *= powmod(order.size() - i, order[i] - last)) %= mod;
last = order[i] + 1;
}
return res;
}
int gondolaSequence[100001];
int replacementSequence[250001];
void solve(){
int i, n, tag;
int nr;
assert(scanf("%d", &tag)==1);
assert(scanf("%d", &n)==1);
for (i = 0; i < n; i++)
assert( scanf("%d", &gondolaSequence[i]) ==1);
switch (tag){
case 1: case 2: case 3:
printf("%d\n", valid(n, gondolaSequence));
break;
case 4: case 5: case 6:
nr = replacement(n, gondolaSequence, replacementSequence);
printf("%d ", nr);
if (nr > 0){
for (i = 0; i < nr - 1; i++)
printf("%d ", replacementSequence[i]);
printf("%d\n", replacementSequence[nr-1]);
}
else
printf("\n");
break;
case 7: case 8: case 9: case 10:
printf("%d\n", countReplacement(n, gondolaSequence));
break;
}
}
// int32_t main(){
// ios_base::sync_with_stdio(0);
// cin.tie(0); cout.tie(0);
// if (ifstream("input.txt").good()){
// freopen("input.txt", "r", stdin);
// freopen("output.txt", "w", stdout);
// }
// int t = 1;
// while (t--){
// solve();
// nl;
// }
// return 0;
// }
